PFC/JA-82-28
Observation of self-binding Turbulent Fluctuations in Simulations
Plasma and there Relevance to Plasma Kinetic Theories
Robert H. Berman, David J. Tetreault, and Thomas H.Dupree
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139
Observation of self-binding turbulent fluctuations in simulation plasma
and their relevance to plasma kinetic theories
Robert H. Berman, David J. Tetreault, and Thomas H. Dupree
MassachusettsInstitute of Technology, Cambridge,Massachusetts02139
(Received 24 November 1982; accepted 21 April 1983)
Non-wave-like fluctuations of the phase-space density are observed in simulations of turbulent
plasma. During decay from an initial state, the mean square fluctuation level decays at a much
slower rate than that of an individual fluctuation. The distribution function of the fluctuation
amplitudes becomes non-Gaussian (skewed) in favor of negative fluctuations. An enhancement in
the aggregate fluctuation lifetime is also observed when the turbulence is driven by an external
source. A model based on a collection of self-binding negative fluctuations, called phase-space
density holes, can explain the observations. Collisions between holes produce hole fragments and
lead to fluctuation decay. However, the hole fragments are self-binding and tend to recombine
into new holes. The implications of these results for kinetic theories of plasma turbulence are
discussed. In particular, it is shown that the theory of clumps, when suitably modified to include
fluctuation self-binding, can explain many features of the nonlinear instability recently observed
in computer simulations.
I. INTRODUCTION
We present a series of numerical simulations which
were designed to test specific aspects of kinetic theories of
plasma turbulence and to probe the behavior of fluctuations
moving at speeds less than the thermal velocity (i.e., nonwave-like phenomena). The simulations are of a one-dimensional, one-species plasma. Besides the reason of simplicity,
we have focused on the one-dimensional problem because
the large number of particles required for phase-space diagnostics would be prohibitive in a three-dimensional simulation with present-generation computers. However, we believe the phenomena we report here will occur in
three-dimensional plasma with a magnetic field. The simulation results are both novel and surprising, especially in the
light of the conventional wisdom that small-amplitude fluctuations in a one-dimensional, one-species plasma have been
thoroughly studied and understood with both simulation
and theory of the past twenty years. Apparently, the reasons
the novel phenomenon we discuss has been overlooked for so
long are because of the historical preoccupation with waves
and preconceived ideas as to what phenomena are important. The simulations discussed here give compelling evidence for the existence and importance of non-wave-like
fluctuations in turbulent plasma. These fluctuations are
small-scale (on the order of a Debye length and a tenth of the
thermal velocity in size), self-binding depressions (holes) in
the phase-space density rather than waves. The turbulence
can be understood in terms of a superposition of these phasespace holes rather than a superposition of waves. This state
of affairs is reminiscent of fluid turbulence where the turbulence is described as a superposition of eddies (vortices). It is
noteworthy that the importance of nonlinear, non-wave-like
phenomena in fluids has been known for decades.
Early efforts at deriving a renormalized theory of plasma turbulence involved a one-point description or theory of
the coherent response.1-3 This produces a model in which
the turbulent plasma is described by a nonlinear dielectric
-
2437
Phys. Fluids 26 (9), September 1983
function. However, further work shows that it is essential to
derive a renormalized two-point equation (correlation function equation) in order to include an important class of nonlinear random fluctuations called clumps.'" These random
fluctuations result from a fine scale granulation of the plasma phase-space density. Clumps can be enhancements
(6f>0) or depletions (bf(0)in the phase-space density f The
depletions have been referred to as phase-space holes.' Here
5f=f - (f), where (f) is the ensemble-averaged phasespace density. This type of fluctuation appears to be an important and prevalent component of plasma turbulence and
would appear to be present in any plasma containing turbulent transport processes. For example, recent work has
shown that clump fluctuations can grow in amplitude (are
unstable) in a variety of linearly stable plasma configurations. 9 The renormalized analytical theories of these fluctuations are necessarily approximate and it has been argued
that they fail to describe an important aspect of the dynamics
of these fluctuations, namely the self-interaction or tendency
of some of these fluctuations to form into bound states.5 In
this paper, we present simulation results that appear to confirm the existence of these self-bound fluctuations in turbulent plasma. These results have important implications for
the clump instability as well as plasma kinetic theory in general.
We have devised novel diagnostics in order to investigate the self-interaction feature of the fluctuations. In the
past, both theory and simulations of plasma turbulence have
focused on the mean square fluctuation level. Conventional
thinking has held that when the distribution of fluctuation
amplitudes is sufficiently Gaussian, the fluctuation correlation function is adequate to describe the turbulence. However, the diagnostics show that the self-binding fluctuations-because they are negative and have enhanced
lifetimes-produce a pronounced skewness to the fluctuation distribution.
At present, renormalized kinetic theories-which are
0031-9171 /83/092437-23$01.90
@ 1983 American
Institute
of Physics
2437
essentially perturbative in the electric field-do not describe
the self-binding or particle-trapping tendency of the fluctuations. This self-binding feature is responsible for the tendency of the negative fluctuations to coalesce into new (negative) fluctuations. In the absence of an external source (i.e.,
the system decays from an initial state), the coalesced fluctuations become progressively more isolated from each other
in phase space as time evolves. We have observed this reduction in the density of the fluctuations in phase space-sometimes referred to as intermittency-in the simulations. It is
difficult to see how kinetic theories, which assume that fluctuations are uniformly distributed in phase space (rather
than concentrated in local regions), can treat this intermittent feature of the fluctuations.
In addition to these general implications for kinetic theories, the one-species simulation results described in this paper have direct relevance to the clump instability in a twospecies plasma.- The enhanced lifetime of the self-bound
clumps that we observe in the simulation implies that the
threshold for the clump instability is lower than that previously calculated.' The magnitude of the enhancement observed here (approximately a factor of 3) brings the theory of
the clump instability threshold into agreement with observa6
tion.
At present, renormalized theories of plasma fluctuations such as clumps focus on the two-point fluctuation
correlation function (6f(l)bf(2)) = (f6(x1 ,v1,t )8f(x
2 ,v2 ,t)) For detailed discussions of these theories, the reader is referred to Ref. 4 and the review article of Ref. 10. These theories derive a generic evolution equation for (6f(I)bf(2)) which
can be written as
(1)
T, ) (&f(l)8f(2)) = S,
(2)) so that Eq. (1) is
where T 1 2 andS depend only on (8f(
indifferent to the sign of bf. Here S is a source term of the
clumps and arises from the rearrangement of regions of different phase-space density by the turbulence. In its simplest
form, the operator T 12 describes the destruction of the fluctuations by velocity dispersion and diffusion [as in Eq. (7)].
In such a theory, the fluctuations decay because the orbits of
their constituent particles undergo stochastic instability.3
The operator T 12 determines the characteristic time
r., (x _,v - ) for two particles to diverge stochastically given
that their initial separations were x_ = X1 - X2,
v =vI -v 2 . Consequently, when S=0 in Eq. (1), the
phase-space density will be torn up into smaller and smaller
grains as time elapses: (6f(l)bf(2)) =0 for t>r-,(x_,v _).For
S #0, this destruction process is compensated by the production of new fluctuations. In the steady state, (6f(l)&f(2))
re,(x_,v_)S.
In order to test Eq. (1), we measured various characteristics of the fluctuations for both S= 0 and S #0. The S= 0
case, which we refer to as decay turbulence, involved the
decay of an initial distribution of fluctuations. In the S #0
experiments, which we refer to as driven turbulence, we generated fluctuations by imposing a given external electric field
spectrum on the plasma. The fluctuation correlation function was measured at equal time intervals during the experiments. We also followed particle orbits in time in order to
\t Id+
2438
Phys. Fluids, Vol. 26, No. 9, September 1983
test the 7 ,(x_,v_) model of T1 2. In addition, we investigated
one of the basic assumptions of Eq. (1), i.e., the fluctuation
amplitudes have a Gaussian distribution. This assumption
implies that fluctuations with bf(O are equally as probable
as those with 6fy0. In order to test this, we divided the phase
space into cells of size 4x. by A v. and made a histogram of
the average fluctuation Wf~ found in each cell. We denote the
probability density of finding a fluctuation 6f in a cell as
P( 6f).
The principal results of these measurements can be stated as follows. In the decay experiments [where S= 0 in Eq.
(1)] we found:
Al. The correlation function does not decay as Eq. (1)
predicts: (6f(1)tf(2)) #0 for t>,ri,(x _,v _).
A2. The characteristic velocity and spatial widths of the
correlation function increase with time while (bf(1)2) decreases with time.
A3. The particle orbits undergo stochastic instability in
agreement with the r, (x_,v.) model.
A4. P( bf ) becomes non-Gaussian (skewed) in favor of
fluctuations with 6f(0. The skewness (Sf(1)3)/((8f(1)2)
3
1.
A5. P( f) becomes more skewed as time elapses.
A6. The skewness decreases with increasing Ax. and
A4v [the phase-space cell used to measure P( 6!)].
In the driven experiments where S= S ext 0 in Eq. (1),
we found:
B1. (&f(l)f(2))~3rc,(x _,v _) S"', where Eq. (1) predicts (bf(l)6f(2)) = rj,(x-,v-) S".
B2. P ( 6!) is Gaussian.
was typically of order -
Item A4 is probably the most striking evidence of defi-
ciencies in the existing kinetic theories of plasma turbulence.
It implies that during decay turbulence, the sytsem is composed of a few deep phase-space "holes" (bf is large and
negative) with bf > 0 (but small) between the holes. To see
this, consider a system composed of a large number of identi-
cal holes-each with amplitude Sf- (0 and phase-space area
A = Ax A v. Let bf, denote the amplitude of 6fbetween the
holes and let the fraction of phase-space area occupied by
holes (the hole-packing fraction) be denoted by p. Then,
charge conservation implies
p-6f- + ( -p) bf, = 0
(2)
bf - /bf+ = (P - 1)/p .
(3)
or
Now consider P( bf) for this simple system with a phasespace cell or window of area A. = Ax., Av.. The quantity
P ( bf) will be significantly skewed toward f(0 if two conditions are satisfied. First, we must have - f->8f, which
implies that the packing fraction must be small (p<j). Second, the average number of holes (R) in the window must be
small. If i> I and the holes are randomly located in phase
space, then P( bf) would be Gaussian. Therefore, we must
have p <I and pA, <A (see A6) in order to have significant
skewness.
We believe that the bf(0 fluctuations implied by the
skewed P( bf)'s that we observe are related to so-called
Berman, Tetreault, and Dupree
2438
phase-space density holes.'" A single isolated phase-space
hole is a self-bound equilibrium which behaves like a selfgravitating fluid element. This equilibrium is characterized
by a definite relationship between its amplitude (f) and its
phase-space dimensions (lx,Av). Such a hole is a state of
maximum entropy subject to constant mass, momentum,
and energy. It is a fluctuation in the phase-space density for
which the potential energy fluctuation is negative and of the
order of the kinetic energy fluctuation. It is a BernsteinGreene-Kruskal mode.
The binary interaction between two equilibrated phasespace holes has been investigated in Refs. 5 and 11. It has
been shown that the two holes (with Ax less than a Debye
length and small relative velocity) will attract each other and
coalesce into a new hole oflarger phase-space dimensions (cf.
Fig. 6 of Ref. 11). In doing so, the average- or coarse-grained
fluctuation level is reduced because some of the original hole
material gets "mixed" with the interstitial background material (6f>0) between the holes. Collisions between two
equilibrated holes with larger relative velocity leads to tidal
deformations of the holes and the production of hole fragments (cf. Fig. 7 of Ref. 11). In a turbulent plasma with many
phase-space "holes," we would expect similar processes to
occur. Indeed, the entropy arguments of Ref. 5 imply that
any group of fluctuations with 8bf0 would tend to form into
a collection of phase-space density holes. Any fluctuations
with 6f >0-being local enhancements in the phase-space
density-would tend to blow apart and form the interstitial
background between holes. Because of the random interactions of a turbulent plasma, a 6f(0 fluctuation will only approximate an equilibrium (Bernstein-Greene-Kruskal)
phase-space hole. In a turbulent plasma, a phase-space density hole retains its tendency to self-bind but, because of collisions with other holes, it never equilibrates (virializes). Any
hole fragments produced in these collisions will subsequently tend to recombine (coalesce) with other holes. The process
of collision and recombination of holes is an essential feature
of the turbulent state.
A simple physical model-based on the interactions of
a collection of holes-can apparently explain the essential
features of the simulation results. Consider the decay experiments. Any given initial distribution of fluctuations will tend
to form into a collection of these holes and cf>0 background
material. The fluctuation level will decay as collisions
between holes produce hole fragments which then mix into
the background. For p -, these turbulent collisions occur
on the re,(x_,v_) time scale so that an individual hole lasts
only for a time r.(x _,v-) (cf. A3). However, the hole fragments produced during these collisions will, because of their
self-binding feature, tend to recombine (coalesce) into new
holes. Therefore, the mean square fluctuation level decays at
a much slower rate than that predicted by the stochastic
instability model (cf. Al). The correlation function will increase in width even as (bf(1)2) decays in time because hole
coalescence will produce holes with larger scale lengths (cfd
A2). Consequently, the holes with small scale lengths will
decrease in number as they coalesce into the new (largerscale-length) hole structures. This reduction in the packing
fraction of the (smaller) holes produces a skewed P ( f ) (cf.
2439
Phys. Fluids, Vol. 26, No. 9, September 1983
A4). As time elapses, these processes will continually reduce
the hole-packing fraction so that P( 6]) will become more
skewed (cf. A5). This decay of the mean square fluctuation
level ((bf 2 )) can be described by the following model equation for the hole-packing fraction p:
dln(bf 2 ) _ d Inp
dt
dt
_
2pAv
bAx
(4)
Equation (4) assumes that the net effect of hole collisions and
hole fragment coalesence is to keep hole size and depth constant while the number of holes decreases with time. The last
term on the right if Eq. (4) is the aggregate fluctuation (hole)
decay rate. If the holes were closely packed in phase space
(p - ), then the hole-hole collision rate would be approximately A v/Ax -r7' where A v and Ax are the hole dimensions. However, for p~4, collisions between holes will be less
frequent because the holes will be more isolated from each
other in phase space. The factor of b > 1 in Eq. (4) accounts
for the fact that hole recombination will reduce the aggregate
hole decay rate below the hole-hole collision rate. The simulation results imply that b 3. In the case of the driven experiments, the packing fraction (and (6f')) does not decay
with time because the externally imposed waves are continually creating holes (clumps)by rearranging the phase-space
density (cf. B2). The fluctuation amplitude produced by this
rearrangement is larger than that predicted by Eq. (1), since
the hole self-binding effect not included in T,2 will increase
the fluctuation lifetime (cf. B 1).
We designed the experiments to simulate the features
typical of turbulent plasma. In the decay experiments, turbulence developed from an initial state of local depletions
and enhancements of the phase-space density. The initial
fluctuations were distributed throughout the phase space
and were of size Ax = A D byA v = 0. lu (A D and vih denote
the Debye length and thermal velocity, respectively). Figure
1 c) shows the P( 6]) that we observed at the end of the
simulation when cop t = 220(w, = Vi, /AD is the plasma frequency). The Gaussian curve (dashed line) in Fig. 1(c) represents P ( 6f]) if the fluctuation amplitudes had a Gaussian
distribution. The skewed "tail" of P ( 8]'), where eSf <0, is
due to the remnants of deep holes initially present. The shifted peak ofP ( f) for8f/>0 is due to the background material
that rapidly decayed (from bf/(f) = 1) as it blew itselfapart
by charge repulsion. The P( 8f) extends further to the left
than to the right because the fluctuations with bf60 decay
more slowly than those with b)1>0. The hole fluctuations
decay more slowly because of their tendency to self-bind and
recombine with other holes. The correlation function at
w, t = 220 [Figs. 1(a) and 1(b)] shows the existence of fluctuations with phase-space dimensions of the same order as
those initially present. However, we show in Sec. IV that the
(x_,v_) model predicts that no fluctuations with sizes
larger than Ax, = 0.06 1A D by Av, = 0.005u, (the smallest
measurement cell) should exists after w, t = 60.
ir,
At a qualitative level, interpreting the fluctuations as a
superposition of holes seems to provide a simple explanation
of the simulation results. In Sec. VI we attempt to make a
more quantitive comparison by deriving a formula relating
Berman, Tetreault, and Dupree
2439
,
,
o.oerF
ao
++
%0
-++
+
0
+++/
00
+
-0.02
Q10
0.00
0.10
V.JVth
0.06
.b
XX
-1.5
0.0
X-./XD
NO>
Z.
-0.021.5
10-
-0.15
_0.00
0.10
FIG. 1. The two-point fluctuation correlation function C(x ,v_) for phasespace separations x_ =x, - x2 and v- = v- - v2 and the probability
P( 6f) of observing a fluctuation bf in a phase-space cell: (a) C(0,v _); (b)
C(x _,0); and (c) P( 8f) at ,t = 220. In this, and subsequent pictures of
P( bf), the dashed line is a Gaussian P( 6) of equivalent width.
P( 6f) to the distribution of Bernstein-Greene-Kruskallike holes. This has proven to be a difficult task for two reasons. First, one must know the structure of an individual
hole in the turbulent environment of many other holes. Second, the coalesence and destruction of holes produces new
holes with different phase-space dimensions and leads to a
distribution of hole sizes. Using a plausible model for the
hole distribution in phase space, we solve a model equation
for P ( 6f ) and calculate various integral properties ofP ( /)f
such as f ,,,d 6! (6f)' P( bf). Analyzing these derived
quantities such as the correlation function (m = 2) and the
skewness (m = 3) in relationship to the other measurements,
we show that much of the simulation results are consistent
with the model distribution of Bernstein-Greene-Kruskallike holes.
The hole model is reasonably successful in explaining
the simulation results even though we have made simplifying
assumptions pertaining to both the hole structure in a turbulent plasma and to the phase-space distribution of the holes.
Efforts to make the model more realistic have led to models
that are prohibitively complicated. We have therefore focused on a simplified model that appears to contain some of
the essential features of the problem. Throughout the paper
we have made a conscious effort to separate the simulation
2440
Phys. Fluids, Vol. 26, No. 9, September 1983
results from the hole model used to interpret them. We believe that the simulation results (Sec. III) stand independently. However, we also believe that the hole model and the
conclusions inferred from it, are a meaningful first step toward a more complete analytical theory of the self-binding
effects.
It is clear that the existing kinetic theories of plasma
turbulence are inadequate to explain the simulation results.
We believe that there are several salient features of our findings that have particular significance for these theories.
1. The hole (bf<O) fluctuations appear to dominate the
turbulence. In the decay turbulence simulations, the correlation function decays at a slower rate than Eq. (7) predicts.
This is clearly due to the mutual attraction and coalescing of
hole material rather than the repulsive tendency of the ff >0
background material. However, Eq. (1) treats 6f>0 and
6f<O fluctuations on an equal footing. This indifference of
Eq. (1) to the sign of 6f implies a distribution of fluctuation
amplitudes, P( V), that is Gaussian. However, for packing
fractions less than 1, P( Sf) is not Gaussian. In this case, the
fluctuations are concentrated in isolated local regions-a situation referred to in fluid turbulence as intermittency. " The
neglect of higher-order correlation functions, which is the
basis of Eq. (1), is generally thought to be valid only when
P( 6f) is approximately Gaussian. " However, it is difficult
to see how any theory which depends only on (8f f) can
describe fluctuation self-trapping, which intrinsically depends on the sign of 6f.
2. An individual fluctuation undergoes stochastic instability and, therefore, decays at the rate -rI(x_,v_)- for
p = j or pA v/Ax for p < . However, the aggregate fluctuation level decays at a much slower rate because the holepacking fraction decreases with time and the holes tend to
recombine into new holes. This is reminiscent of fluid turbulence where the aggregate fluctuation level decays at a
slower rate than that for the dispersion of a passive scalar
contaminant." The existing theories of T12 [e.g., Eq. (7)]
treat orbit stochasticity but neglect self-binding and thus
predict that the individual and aggregate fluctuation levels
decay at the same rate.
3. The trapping tendency of particles which leads to the
self-binding tendency of the holes is an important feature of
the turbulence. In the existing theories, it is assumed that
trapping can be ignored if the electric field correlation time
r., is less than the trapping time r,. T12 is then constructed
by a renormalized perturbation expansion in the electric
fields. Since such expansions use unperturbed orbits as the
lowest-order approximation, it does not seem possible to describe particles trapped in holes by these conventional expansion techniques.
Though a proper theory of these effects is lacking, the
simulation results discussed in this paper give an indication
of the modifications needed in existing theories. For example, consider the recently reported nonlinear instability in a
one-dimensional simulation ion-electron plasma.' The instability has been identified with the electron-ion clump instability.' However, the observed threshold for the onset of
the instability is lower than that calculated from the existing
theory of the electron-ion clump instability. The discrepanBerman, Tetreault, and
Dupree
2440
cy is resolved when we include the effect of hole self-bind
in the clump theory: the instability occurs at a lower thre
old than Eq. (7) would predict since the tendency of the fl
tuations to self-bind enhances their lifetime and, thereft
their ability to regenerate. We consider this effect quant
tively in Sec. VIII and show that the results of the simi
tions can be used to bring the clump theory into approxim
agreement with the observations of Ref. 6. Schematica
the result can be seen intuitively as follows. Clumps of phg
space density are unstable when their source S in Eq.
overcomes their decay rate T 12 . Since S is proportional
(6f(l)5f(2)) (cf.Sec. II), we can define the operator r so t
hat
S = r (6f(l)6f(2)). Then, with the replacements T,2 -t*i.i
and 2r-*d/1dt in Eq. (1), the clump instability growth rate y
satisfies the schematic relation'
(5)
where M = rjTC. In Eq. (5), the M term is due to the clu mp
source term S, whereas the - 1 term is due to the deca3 of
clumps by stochastic instability (T 12-+-o; 1). However, wiien
p~1, the simulations imply that the mean square fluc:tuation level decays at a slower rate than ri , i.e., the simi lations imply that r,7' should be replaced by (bre
1 )~ wh ere
b ::3. The factor b models the tendency of the fluctuati ons
(holes) to recombine [cf. Eq. (4)]. Since M is proportionalI to
-r,, Eq. (5) becomes
y~(1/2rl) (M - 1/b)
in the presence of self-binding. Therefore, if b-3, a lov
threshold for the instability would result since 9 -0.3
quires a smaller free energy source (S~M) for instabil
than indicated by 9 - 1.
'
D_
a_
G (x_,V-) =S,
(7)
G(x,v-)= (of(l)Sf(2)) and x. =x 1 -x 2, v_
v2 are the phase-space separations of the two points
= V1 (1) = x1,v, and (2) = x 2 ,v 2 . The velocity-space diffusion coefficient D_(x-) describes the relative diffusion between the
two phase-space points. The source term takes the form
where
S= 2 (D 12 (x-) d(f(l)) d (f(2))
\
3v,
-
F 12 (x-) C0 )))
dV2
av,
(8
(8)
In the limit x.-+0, the functions D 12 (x_) and F12 (x_) be2441
Phys. Fluids, Vol. 26, No. 9, September 1983
/
(f(2))\
d,2
(9)
where D2
ix(x-) is defined by
D
et
2
f
do (E 2 (k,w))*
- - 2_ dk
i J 2r J 21r
Ie(k,w)12
X rb(w - kv) exp(ikx-) .
(10)
2
Here, (E (k,w))*e" is the external electric field spectrum for
wavenumbers k and frequency w, and e(k,o) is the plasma
dielectric function that shields the external fields.
The decay of the fluctuations described by the left-hand
side of Eq. (7) is due to the random increase in the relative
separation of particle orbits. For small separations,
D_ = k 2 Dx2 ->0 so that two particles diffuse together
since they feel approximately the same forces. Here D is the
one-point velocity diffusion coefficient of quasilinear theory
and ko is an average wavenumber characteristic of the turbulence,
1
( 2D_
(11)
02D (_x2D_=
Kinetic theories of plasma turbulence assume that
plasma phase space is completely filled with fluctuations.
terms of the bf(O (hole) fluctuations, this means that
hole-packing fraction is 1. In their simplest form, the theoi
describe the decay of turbulent fluctuations by balli!
streaming and diffusion.3 A fluctuation will be shea
apart in space by the velocity dispersion of its constitu
particles and diffused apart in velocity by turbulent elect
fields of other fluctuations. This process of stochastic ox
instability is described by an equation of the form [cf.
(1)],
--
d
(x i d 3(f(s))
wherDS=
D f~ _ i av,
2
11. REVIEW OF RENORMALIZED KINETIC THEORIES
+ _
come D and F, where D and Fare single-point diffusion and
dynamical drag coefficients. 4 In a one-species plasma, as is
the case here, the D and F terms due to the plasma selfconsistent fields cancel exactly. However, S can be nonzero
in a two-species plasma or if an external source of electric
field fluctuations is imposed on the plasma. For the case of
externally applied fields,
If two particles have large separations, D_-*2D since they
feel different forces and diffuse independently. The smaller
the initial separations, the larger is the relative orbit diffusion time. Let us define re(x_,v_)as the time for two phasespace points initially separated by x_,v- to be separated
spatially by k '. The ensemble-averaged orbits defined by
the operator T,2 of Eq. (7) imply
- (xt ) = 4(D) =4kD
(x2_,
(12)
for kx - I < 1. For steady-state turbulence, the time asymptotic solution of Eq. (12) is
(x 2 (t)) =
I
(x 2
-
2xv-,ro + 2v2 r2) exp(t/ro),
(13)
where
(14)
(4k D) 11 = (12)2"1 r..
The particle orbits described by Eq. (12) diverge exponentially until t = rl (x-,v_), where
=
-r,,(x ,v
rIn
(
2
-2xro+2v2X)3
(15)
when the argument of the logarithm is larger than unity and
r*' = 0 otherwise. For t>r,,(x-,v_), the orbits diffuse independently.
In the decay turbulence experiments, we will focus on
Eq. (7) with S = 0. In these experiments, G (x_,v_) and D
decay from an initial state. However, because the time dethe
pendence ofD appears only weakly in ro = (4k 2D)-13,
Berrnan, Tetreault, and Dupree
2441
steady-state expression for r(x_,v_) given by Eq. (15) is still
Here R is defined by
approximately the characteristic time for an individual fluctuation to decay. With this in mind, we can easily obtain the
decay of an initial correlation G (x_,v_,0) when S = 0. We
can express G (x_,v_,0) in terms of the time-reversed orbits
R=fdkkI
2
A (k) [ImE(k,kv)1
|e(k,kv) + | k | [Im e(k,kv)] 2 /(rko)
where
x_ - t) and v_( - t) of Eq. (7) and obtain
G(x_,v_,t = G [x _( - t),v_( - t),O] ,
where x_(0)= x-
and v_(0) = v-.
(16)
Let us assume that
G (x_,v_,0) = 0 for fkox_4>1. Since two particles (initially
separated by x -,v-) will be correlated only for t<r,1 (x_ ,v_),
Eq. (16) implies that G(x_,v_,t) = 0 for t>re,(x._,v_), i.e.,
fluctuations will be "chopped up" from the outside (x_,v_
large) toward the inside (x_,v- small) as time elapses.
For the driven experiments, we consider Eq. (7) with an
externally applied electric field spectrum. Using Eqs. (9) and
(10), we can integrate Eq. (7) along the two-point orbits to
obtain approximately
Im efk,kv)=
r-
0(f)
k2
(25)
3V
is the imaginary part of the dielectric e(k,w) and
A (k) =
dv
S2r I
f dx_
e -"-r,
(x_,v_)
61/2k
D T = D ind + D ext = ( - R)Dex t .
rc,(x_,v_)--oo asx_ and v_--O [cf. Eq. (15)]. The average
phase-space density (f(v)) at a point v will differ from the
average density of a clump of size x_,v _, since (on the aver-
age), the clump carries the average density (f(v - bv)) of its
point of origin at v - v at a time rl (x-,v -) earlier. Therefore, the amplitude of these fluctuations is given by
_,5v)
(f(v))] 2 ) .
-
(18)
Since the particles are diffusing in velocity, (8V) 2
= 21-,(x_,v_)D"*. Assuming that bv d ln(f(v))/Av41, Eq.
(18) yields the result Eq. (17). The-fluctuation level will produce a plasma electric field spectrum due to internal charge
given by
(E 2 (k))
2
dx
Ie(k,wjr Ik I fx4
Xexp( - ikx_) G(x_,v_),
=T-)
(19)
where
G(x_,v_)
G(x_,v_)
=
-
(x_,v.).
(20)
U (x _,v) follows from Eq. (17), but withD-(x) replaced by
2D,
=
U(x -,v._=)
2d(f(1)) d(f(2))
v,
02 ,
(21)
where ~(v)= 2r,,.[4 + (v kor,,)2 1. The plasma diffusion
coefficient induced by D"Y is
D
n"=
-
2
k
2M
d
27r
(E 2 (k,w)) rb(w - kv) .
(22)
Using Eqs. (16)-(2 1), this becomes
D in = RD
2442
Phys. Fluids, Vol. 26, No. 9, September 1983
(27)
3
(17)
whereD ex= D W(0). This solution can be understood in the
following way. The diffusion coefficient Eq. (10) due to the
external turbulence will rearrange the phase-space density,
f(x,v), in a chaotic fashion. However, because of the Vlasov
equation f(x,v) must remain constant along particle orbits. A
small clump of plasma with dimensions x_,v_ will preserve
its phase-space density for a time re,(x_,v_). Obviously,
(6f 2 ) = ([f(V
(26)
(J is in an ordinary Bessel function and the factor 62 is
approximate). The total plasma diffusion coefficient is then
The parameter R is less than unity in a stable plasma.
G(x_,v-)=2rc,(x_,v-)DeXtd(f*()) df2),
dv,
dV2
(24)
2
(231
MII.
DIAGNOSTICS
In the plasma simulation, we treated a periodic system
of length L = 10irA D (A D is the Debye length) in which we
integrate N, particle trajectories. The trajectories were obtained by using a finite time step of 0.2w,- ' (wP is the plasma
frequency) and by solving Poisson's equation on a grid divided into 512 zones. The value of N, A/L = 65 190 was necessary for an accurate determination of the one-particle distribution function and the two-point correlation function
and closely approximates a collisionless plasma. We measured the single-particle distribution function by counting
the number of particles in a phase-space cell. At v = 0 in an
initial Maxwellian distribution, the average number of particles in the smallest cell of size L /512 by v,4 /200 was 8.
Throughout the calculations, the spatially averaged distribution function remained Maxwellian (i.e., fo(v)
2
= (2vh)-1/2 exp
/(2V2)]). Therefore, we generally
achieved an ensemble-averaged measurement by averaging
over the system length. However, for the two-particle correlation function, we performed additional averaging over the
velocity region |vl v, , where the turbulence was assumed
to be homogeneous in velocity. We found that statistical accuracy was not reliable when using small numbers of particles or small numbers of phase-spgce cells for ensemble averaging. When NP 2 /L or the number of particles per cell is
small, integrals of the correlation function can be measured
instead."
During the experiments, we performed a number of
diagnostics on the system every 20wo; ' (up to 220w; 1). To
do so, we divided the phase space into cells of size 4x. by
A v. and counted the number of particles per cell, N. We
then calculated the mean number of particles per cell, (N ),
and the fluctuation about the mean, 6N = N - (N). The
particle distribution (averaged over a cell) is f= NI
(4xAv) and the fluctuation is &f = 6N/(AxAv.). The
bar indicates the average over the cell size.
In order to measure the two-point correlation function
as accurately as possible, we used a phase-space cell of size
4x, =.dx, = L /512 and Av. = v, = vh/200. Let (ij)deBerman, Tetreault, and Dupree
2442
note the coordinates of this cell in the phase space. Then, the
correlation function we measured is
(Wf()
f(2)) =IYY
Sf (iAx.+xj4v
measured the discrete particle distribution function to be
3262(f)
A
15 ()Yf())h ADv,no (d+
+vx
exp(
-
3262
3262(f)A
x f (ifx,j
-,
(28)
where the i sum is over the system length (Iix. I<L ),an d the
j sum is over the small region (A v,) of velocity space where
Av. In (f)/dvI(I|jAvI<32h/200), and M, is the total
number of cells averaged over.
We determined the probability distribution P( 3 f) of
finding a fluctuation Sf = &N/(Ax.Av,) in a phase-space
cell or window of size Ax, by Av.. This was done by ma king
a histogram of the particle number fluctuation, 6N, fouiid in
each window. The phase-space window sizes varied according to Ax, <Ax.< 3A D A
< Av h.The fluctual:ions
v,,
described by P ( f) satisfy
fd
(29)
fP( f)= I
and overall charge neutrality
fd
(30)
f WfP( f)=0
throughout the experiments.
As a further diagnostic, we followed particle orbi ts in
both the decay and driven turbulence experiments. Wieselected M, particles with velocities |v(t0) I<0.Ivh at tim e to.
Tracking these particles in time, we constructed the niean
square velocity change
(4dv 2 )
M
[v(t) -v,(to)]
2
(31)
,
where v,(t) is the velocity of the ith particle at time t. We also
followed the relative velocity between two particles. Let Mbe the number of particle pairs that have v,(t0 ) - )2(t
0)
= V_ ± 0.00 2vh and x,(t) - x 2 (tO) = x ± 0.12
at time
to. Tracking these pairs in time, we constructed the niean
square relative velocity change
(4v_(x_,v_
--M-I
2
)
[v 2 (t ) - v 2(0 )I -
[v,(t) - v(to)]] 2 , (32)
where the sum is over the M_ pairs.
In all the experiments, we are interested in fluctuat ions
in addition to the thermal (discrete-particle) level. Accon ding
to discrete-particle fluctuation theory, the correlation fiunction of a thermal plasma is
(&f(I).5f(2)),h = no- '_(x)(v_)(f(l))
+ (f(l))(f(2))
~n2 D,
(33)
2n04D
where no is the particle number density N, /L. The first t erm
in Eq. (33) is the self-correlation of discrete particles and the
second term is due to the shielding of the discrete particle s by
the spatially averaged plasma distribution (f). In the si mulation, we could not resolve scales less than the smallest nieasurement cell (Ax, = 10irAD /512,4d v, = vh /200) so that we
2443
Phys. Fluids, Vol. 26, No. 9, September 1983
n
I
D
1x
.
(34)
h
where A = 1 for x _<Ax,, v- <A v, and is zero otherwise.
We neglect the shielding term, since it is beyond the statistical resolution of the small measurement cells. Here (f) remained Maxwellian throughout the simulations, consistent
Subwith a collisional self-heating time of order 10 3 0,-'.
correlation
the
fluctuations,
particle
discrete
the
out
tracting
function we focused on is
C(x_,v_) = (6f(l)6f(2)) - (3262/nOADvfA)f , (35)
where f, is the initial Maxwellian.
Particle discreteness will contribute to the P ( bf) measurements. We can estimate this discreteness contribution as
follows. In thermal equilibrium, the particles will be randomly distributed in a phase-space cell. Therefore,
(ON2 ) = (N), so that the probability of finding a discrete
particle fluctuation 6N in the cell is
PA(N) = (27(N))- 2 exp[ -- N2 /(2(N))] . (36)
In terms of f, we can write this as
2f2
-/
p2
(N)
(37)
T
Therefore, the width of the discrete particle contribution to
P ( bf) will be small for large (N), i.e., large windows. This
also implies that the smaller windows-where the contributions to P( Sf ) will be due almost entirely to discrete particle
fluctuations-will have P( f)'s that are Gaussian.
PhA( 5f)=(N)
exp
IV. OBSERVATIONS
A. Decay turbulence
In the decay experiments, we followed the decay of an
initial phase-space distribution of fluctuations. Because linear plasma waves are not strongly Landau-damped for
wavelengths greater than A D, we prepared an initial phasespace configuration that minimizes the excitation of longwavelength plasma waves. We set up a periodic "checkerboard" pattern of local excesses and depletions of particles in
phase space. The correlation function at t = 0 (Fig. 2) and its
contours [Fig. 3(a)] show the periodicity of the checkerboard. Initially, the holes are phase-space regions devoid of
all particles. The particles removed from these holes were
put randomly in the phase-space regions between the holes.
This explains why initially P( f = -fo) is sharply defined
while P( Tf>0) is randomly distributed about 6f = + fo
[cf. Fig. 2(c)].
As time elapsed, we performed frequent diagn6stics on
the system. All measurements discussed in this section were
made in the velocity region Ivl|lv|
where dfo/v::O (measurements made in the region v -vh yielded similar results).
We found that the statistical properties of the system were
not sensitive to details of the initial conditions. After a few
Berman, Tetreault, and Dupree
2443
+
+A
>!o
+ +
+Q0
+
+
+
0.0 0
- o +i
-+
-+
000
01
v
+
b
X
Kx
K
x
K
('Jo
x
K1
K
K
"
K
S
0
U
++
+
+*
-+
'
k 0'--A~A , is given by Eq. (15). The diffusion coefficient in rT
is
@~ , (38)
D~(e/M)2 (E2 ) , -~((E 2 )/4rrnomv' )2
XX
X
K
K
K
K
K
X
K
K
K
K
-0.8
-1.5
0.0
xX
1.5
0
C
0*
-1.0
0.0
9f /fo
1.5
FIG. 2. Initial phase-space distribution of fluctuations described by the
two-point correlation function C(x_,v_) and the probability distribution of
fluctuations P( 45f) for decay experiments: (a) C(0,v _); (b) C(x_,0); and (c)
P( 6!) at ,t = 0.
tens of plasma times, a fully turbulent, random distribution
of fluctuations evolved. This can be seen in the contour lines
of the correlation function at o, t = 40, where there is only
slight evidence for any remnant of the initial phase-space
periodicity [cf. Fig. 3(c)]. By the time w,t = 60, complete
destruction of the initial checkerboard periodicity is
achieved. Also, by this time, the half-width and height of the
correlation function have decreased significantly (cf.Fig. 4).
New randomly distributed phase-space structures of size
Ax<A D by 4v<0.lv, have formed. The steady increase in
the velocity width of the correlation function for 60
<w, t(220 is evident from Fig. 4. This implies that fluctuations with larger velocity scales are being produced. This
occurs even as the mean square fluctuation level [measured
by the peak of the correlation function C(Ax~v,v)] is decreasing with time (cf. Figs. 4 and 5).
A kinetic theory described by an equation such as Eq.
(7) predicts that the initial checkerboard fluctuations are
torn apart by ballistic motion and diffusion. This is the process of orbit stochastic instability (cf. Sec. II). Two phasespace points in a check will be diffused by the turbulent electric fields caused by the large I8f I initially present. The time
(x_,v_) for two phase-space points (initially separated by
x _,v .) to be diffused from each other spatially by a distance
Trc
2444
Phys. Fluids, Vol. 26, No. 9, September 1983
where (E 2) is the mean square electric field. In thermal equilibrium, (E 2 )/(47rnomv2 ) = (nZA)~-. During the initial
decay, the measured value of C (4x ,,Auv) was approximately
seven times the discrete-particle function level [see Eq. (35)]
We recall from Sec. II that orbit stochasso that r,~~
0 I l-'.
tic instability will destory a fluctuation of size k -' down to
the scales x _,v - in a time re (x -,v _.j Therefore, using Eq.
(15), we calculate that the initial checkerboard fluctuations
(of size A D, 0. 1vh) should be "chopped up" or destroyed
down to the smallest measurement cell (of size 0.061A D,
0.005V,h) in a time r1c(0.061AD, 0.005v, )6 o;'. However, we observe fluctuations of size Ax~
, v0.1vh,
even up to t = 220.
In addition to the fluctuation decay time, the qualitative features of the decay process are in disagreement with
Eq. (7). We have observed that C(x,v _,t )for allx -<0.2A ,
and v _<0.02v,, decayed at the same rate (cf. Fig. 6). The
shape of this inner "core" of C(x_,v_,t )persists through the
run (compare Fig. 1 with Fig. 7). However Eq. (7) implies
that C(x_,v_,t) would decay with a time scale
rc,(x_,v_)(60w,-' (larger x_ or v. decaying faster). The
data of Fig. 6 were not accurate enough to determine the
exact decay rate. However, the decay ofthe core is consistent
with a power law decay (time scale ~ 3 ro) or an exponential
decay (time scale ~lOro). (A model for the time decay is
discussed in Sec. VIII.) We have also observed that, for v values outside the core region (Iv -I>0.2vh), the correlation
function decays (decreases) for w, t<60, but increases thereafter (cf. Fig. 4). This leads to a v- correlation function,
C(0,v-), composed of two distinct v_ regions. This is particularly evident in Fig. 1(c). The inner core (|v- I<0.02vh)
forms by o, tz 60 and persists through the simulation. The
main body (Iv_.If>0.0 2vth) of C(O,v-) expands to larger v _
scales for .,,t>60. This increase in scale size apparently occurs in the v - dimension only and not in the x dimension.
From Fig. 5 we see that C(x_,0) retains its form during
60<wt(220, implying a smooth distribution of spatial
scales whose maximum is approximately A . We can define
a characteristic velocity scale (4 ) and spatial scale (Ax) by
C(0,4,)= C(4,0).FromFig.7weseethatAx/4,, 10w '.
For small x-_ and v-, this appears to be the case for most of
the simulation. However, the production of the v_ tail at
later times leads to 4,/A<10w -'. For example, in the extreme case of Fig. 1, A4/4,:::::=l0w in the core region,
whereas A4/Av <5.3c,-' in the tail region.
The decay of the initial checkerboard pattern can also
be seen in the time dependence of P( Sf). Initially, there is a
peak in P( bf ) at Sf =fo for the background material and
at bf = -f, for the hole material [cf. Fig. 2(c)]. In other
words, the most likely hole fluctuation is bf = -fj and the
most likely background level is Tf = +fo. Figure 8(a)
shows a typical P ( bf) at a later time. The amplitude for
Sf >0 has been reduced significantly. The reduction of the
maximum amplitude for fluctuations with bf <0 is less draBerman, Tetreault, and Dupree
2444
0.10
0.10
0.05
0.05
0.00
0100
-0.05
-0.05
-0.10
-0,10
~C
W)
TIME
T- E2
0.00
*
0.10
0
0
FIG. 3. The contours of the two-point correlation function C(x ,_) at (a) w, t = 0; (b)
w, t = 20; (c) o, t = 40; and (d) o, t = 60
show the transition to a turbulent state. The
x axis is x./A D and the y axis is v_-/v,h.
0
TIME 20.00
I
.2
0.05
0.00
0.00
-0.05
-0.05
-0.10
-0.10
0
a
TIME 40.00
TIMWE 60 .00
matic. During the time co, t<80, we observed that the bf>0
material decays faster than the 6f <0 material. Forw,, t>80,
the maximum f <0 is roughly constant in time, while that
of of >0 decreases.
The contribution of discrete-particle fluctuations to the
observed P ( Sf) is small for all but the smallest windows.
For example, using Eq. (37), we calculate that the width of
the discrete-particle Gaussian contributing to Fig. 8(a) is
8f /fo = 0.015. The width of P (f~) in Fig. 8(a) is substantially larger than this. However, the discrete-particle contribution is not small for all window sizes. The P( Kf) for the
smallest window (Ax,,Av,) was Gaussian and due almost
entirely to discrete-particle fluctuations. Also, as time
elapses and the turbulent fluctuation level approached the
thermal level, discrete-particle fluctuations make a larger
and larger contribution to the observed P( 8f) for the same
window size.
Most of the P( f) curves are non-Gaussian or skewed
toward bf <O. The magnitude of the skewness is frequently
of order - 1. The degree of skewness depends on the window size A. and time. Consider a fixed time (greater than
w, t = 80). As A. is increased from the smallest window of
size A, =A, =6.l36Xl0-5AD h to A.=0.05AD Ut,
P ( 6f) goes from being Gaussian to being skewed. As A,,, is
C-(0,V.-) /f 2
1.0-
GO
C(x_, 0)/f 2
0.020-
20
220
0.0
05
V-./Vth
FIG. 4. The two-point correlation function C(0,v_,t) for the case of decaying fluctuations. Note the growth of the width.
2445
Phys. Fluids, Vol. 26, No. 9, September 1983
220
-1.5
0.0
1.5
FIG. 5. The two-point correlation function C(x_,0, t) for the
case of decaying fluctuations. Note the width of C is approximately A v.
Berman, Tetreault, and Dupree
2445
1O.-
I
I
I
i
I
I
~I
I
I
-
4 -a
Co
a-
N C0
+
0
-0.4
+
+
x
0
0.2
_
+
x
X
xx
xx
+
x
+ +
X
0
-b
X
x
++
xp
+x
220
Wpt
s
+
FIG. 6. Time decay of selected values of the two-point correlation function
C(x .,v,t ):C(0,0,t)( + ),C(0. 125A D,.01 vh,t )(X )andC(0.5A D ,0.4v,5 X*).
The decay rate is consistent with t -1.
+
.
-2.00
4-
220
Wp t
increased further, the skewness decreases, i.e., its absolute
magnitude decreases. This dependence of the measured
skewness on window size is shown in Fig. 9. The skewness is
small for very small A, because the discrete-particle contribution to P ( Tf) becomes dominant in accordance with Eq.
(36). The decrease in the skewness for large A, (where discrete-particle effects becomes negligible) can also be seen
from Figs. 10(b) and 10(c). For a given window size, P( Yf)
becomes more skewed as time elapses. This can be seen by
comparing Fig. 10. Figure 8(b) shows a typical plot of the
measured skewness against time.
Figure (11) shows the results of orbit tracking during
the decay ofthe fluctuations. Measurements were made after
0.18
%
a
0
++
++
FIG. 8. A typical P( bf) measurement shows that phase space is composed
of a few deep f<0 holes in a background of shallow f>0 phase-space
fluid. (a) P( 8f) at w, t = 120 for a typical window 4x /A = 1.22 and
Av/v,, = 0.125; (b) the observed skewness s(t) of P( Sf) for the same window ( + ).The crosses (x (are inferred from the theoretical model discussed
in Sec. VI.
the fully developed turbulent state had emerged (after
wt = 80). It is clear from Fig. 11(a) that the particles diffused in velocity ((Av) varies linearly with t). The mean
square relative velocity change for an initial particle separation of v_ = 0.02v*,, x- = 0. 12A D is shown in Fig. 11(b). Up
to w, t - 80 = 40, the relative diffusion coefficient (D-) is
clearly much less than the one-particle diffusion coefficient
(D )of Fig. 11(a). However,D_ ~ 2D thereafter. For the larger initial separation of x_- = 1.47A D, v_- = 0.02vh, we see
from Fig. 11(c) thatD_ ::: 2D. These results are in agreement
with the stochastic instability model. Recall that k 0-'~i=' D
for the decaying fluctuations. Therefore, D-.(D for
C;)
-+-+
0.00
AW/AD
.0
I9
-0.04
.
-0.10
0.00
I
.
I
.
I
I
I
.
.
I
"i.
0.10
V-/Vth
0.1
+
.
b
+
S
0
I~
-0.04-
+
xX
1.5
0
X-/
1.5
+
+
+
XD
FIG. 7. The two-point correlation function at an intermediate time in the
decay case: (a) C(0,v ) and (b) C(x_,0) at w, t = 120.
2446
03
.
Phys. Fluids, Vol. 26, No. 9, September 1983
FIG. 9. Observed skewness of P(
Ax /Adv::=00,-
f)
decreases for large windows with
Berman, Tetreault, and Dupree
2446
3- a
0
-0.6
0.4
8f/fo
4 b
o
-0.3
97/f
0.2
0
1
10-
FIG. 10. P( Kf) for windows with (a)
Ax/2D =0.6l4 andv/u ,h =O.lat
t = 120; (b) Ax/A D = 2.454 and
Av/vh =0.075 at o,t = 120; and (c)
X/A = 2.454 and Av/v,, =0.075
at , t= 220. The skewness decreases for larger windows [(a) and
(b)] and increases with time in a given
window [(b) and (c)].
C
C:
-0.25
0.00
0.15
' fo
x_ =0.12ArbutD =2D forx- = .47AD.Wealsonote
that rozu,,-' during the decay, so that Eq. (15) gives
re,(x _,v_)~42w)- for initial values of x =0.12AD,
v = 0.02Vh. This is consistent with Fig. 11(b) where the
particles diffuse independently after a time interval
r,1 (x_,v_). We conclude that the particle orbits are undergo-
6
A
CY
V
9
to
240
WlPt
to-\
b
A
V
2, 0
Wpt
FIG. 11. Measurements of a particle's
mean square velocity change ((4 2))
and the mean square relative velocity
change between two particles ((40v ))
for the decay case: (a)normalized (Av 2)
for 80w, t240; (b) normalized (AV2
for initial separation x_1j2D = 0. 12,
v/v,h =0.02;
and (c) normalized
(4V2 ) for initial separation x_/
A = 1.4,vvh =0.02.Thenormalizing factor is v2 /(nA D )2/- and also in Fig.
15.
0
A
nli
V
240
so
WPt
2447
Phys. Fluids, Vol. 26, No. 9, September 1983
ing stochastic instability at the rate r,,(x_,v_)- . An individual fluctuation therefore decays at the rate rl,(x_-,v_ )'.
The decay of an individual fluctuation can also be inferred from the contour lines of constant C (x_-,v.) (cf. Fig.
3). The contours resemble nested ellipses whose major axis is
tilted toward x_,v_ >0. This tilt is consistent with a decaying fluctuation since x - >0 and v >0 regions are correlated.3 Moreover, the angle which the major axis of the contours of Fig. 3 makes with the x _ axis is consistent with that
for stochastic instability. To see this, we note that the contours of constant -re,(x_,v_) can be written as
-x2
X_TD )
-2
2 or+
(41D
(OW;h)
-_
*h
(0. l0
/
(0.
0)
)2 = const.
(39)
For ro=10, these contours approximate the tilted contours of Fig. 3. We conclude from this and the (4v 2 ) measurements that the fluctuations producing the closed contours of Fig. 3 are being torn apart at a rate r,(x_,v_)- 1 .
This rate is faster than the decay rate we have observed for
the aggregate fluctuation level (6f)2 . We believe that this
occurs because an individual fluctuation decays at the rate
rc (x_,v_)-', but also tends to recombine with other fluctuations.
B. Driven turbulence
In this series of experiments, we started with a spatially
homogeneous, Maxwellian distribution of particles. We then
imposed a group of 300 waves with a wide phase velocity
spectrum (4 vP, = 3vt,). In order to minimize transient effects, the waves were turned on adiabatically from t = 0 and
reached constant amplitude by w, t = 60. All the diagnostic
measurements were made after this time. The wavelengths
were chosen randomly between A D and 2A D in order to
avoid mode coupling between the waves and any long-lived
plasma oscillations. The wave amplitudes were chosen such
that their turbulent trapping width2 would =:0. 1v,.
Consistent with the clump theory, fluctuations were not
produced near v = 0 (where dfo/&v=0), but were generated
in the region near v = v,, (where dfo/dv# 0). Figure 12
shows the correlation function generated at v = vt, when the
particle orbits were determined with only the external waves
present (Poisson's equation was not invoked). Figure 13
shows the result when the particle orbits were determined by
the external waves and the plasma self-consistent fields
(Poisson's equation was invoked). The peak values of the
C (x ,v_) were enhanced in the self-consistent case by 1.8
over the non-self-consistent case. A typical P ( f) curve for
the self-consistent case is shown in Fig. 14(a). It is evidently
Gaussian, as were the P( 8!) curves for the non-self-consistent case [cf. Fig. 14(b)].
These results can be directly related to the renormalized
kinetic theory described by Eq. (7). We recall from Sec. II
that an externally applied electric field spectrum of waves,
(E 2 (k,w))*', will randomly rearrange the particle distribution and create clumps of phase-space density. In the absence of the plasma self-consistent fields, the external waves
Berman, Tetreault, and Dupree
2447
*
-a
r-
0.50
0.30-
a+
+
La
+
+
*
No
*
WC)
.'**1-
4q
C
C
-015 0
0
-015
0
D.12
FIG. 12. The two-point fluctuation
correlation function for the driven
case without Poisson's equation: (a)
C(0,v); (b) C(x-,).
V_/Vth
0.3c
-0.12
V-/Vth
T
-
0.50
-
FIG. 13. The two-point fluctuation
correlation function for the driven
case with Poisson's equation: (a)
C(0,v); (b) C(x-,0).
7b
b
K
NO
X
1
.0
X
C-,
0
"It
0.
-
0)
_Q15
-1.5
0
1.5
-1.5
x-/XD
will produce a fluctuation level given by
(bf(1)6f(2)) = 2,ro
03(f(l))
l d(f(2))
Ov( ,
(40)
12 (x- 1
where we have evaluated Eq. (17) at the x_,v_ values where
re,(x_,v_)= ro. As we shall see, this choice ofx-,v_ is convenient for comparisons of the self-consistent and non-selfconsistent cases, i.e., the weakly varying logarithm factor in
-rc,(x_,v_) can be set equal to unity. Moreover, we will focus
on the ratioof Eq. (40)to an analogous self-consistent expression, in which case the.logarithm factor no longer enters. In
Eq. (40), .9 1 2 (x-) is given by
'012(X_) =
kmI
2vr
2 _(E
1r
2(k,w))
and o = (4k22)-'I where 9
=
912(A),
fluctuation level in the presence of .9 and the self-consistent
fields, ie., T 12 on the left-hand side of Eq. (7) is assumed to
include hole self-binding as well as the velocity streaming
and diffusion terms. Since the data gives a value of 1.8 for the
ratio of the left-hand side of Eq. (43) to Eq. (40), we obtain
= 3.96.
Let us now compute the value of r Iro predicted by the
theory of Sec. II in order to compare it with the value 3.96.
According to Eq. (7), -r.,,can be determined from Eq. (11) and
Eq. (14) by using D_ due to the total (external and induced)
fluctuation field [cf. Eq. (27)]. Therefore, for small x-_,
t
D._ =(k|D"+kDext)x2_ =(k 2R + k )Dexx2
=(k
Xirb( - kv) exp(ikx_),
(41)
and k, charac-
3
k2 _
1
.9
(42)
"
29 \ ax2
Estimates of ro and . show that Eq. (40) reasonably approximates the fluctuation level of Fig. 13. In the presence of
the plasma self-consistent fields, .0 in Eq. (41) will be reduced by the plasma dielectric, i.e., the external waves will
be shielded by the plasma. For measurements at v = v,
and Im e = 0.76(kA
D
2.
-3v
I
2(X-)
(f( 1))
Ofl
1
d(f(2))
'3V2))
dv2
(43)
where r., is the characteristics lifetime of the mean square
2448
Phys. Fluids, Vol. 26, No. 9, September 1983
F7i'
U~
-0.30
0
0.30
g-f/f 0
We take
k=A ' since the wavelengths of the driven fields are
weighted more heavily in D "t (we have verified this in the
orbit tracking experiments). Therefore, El2 = 2.2 and
= 9/2.2. Then, according to Eq. (1), the fluctuation level due to 2 and the plasma self-consistent fields is
(6f(I)bf(2)) = 2S(2.2)~'.1
R + kn )|eF-2TX2
(44)
where k S is derived from the induced fields [D_ = D " in
terizes the driven fields, i.e.,
Re e= 1 + 0.28(k" D)-2
1.5
0
X-/X./
b
a.
-1
-020
0
0.25
9f/f 0
FIG. 14. P( f) with Ax/AD = 2.454, Av/V = 0.1 for (a) non-self-consistent driven case; (b) self-consistent driven case.
Berman, Tetreault, and Dupree
2448
Eq. (11)]. Therefore, the stochastic instability model foi
would give
- -
2
2
1/
T12
.
(45)
(k n
el
From Figs. (12) and (13), we find that k,/k,0.67 so that
Eq. (45) implies that r~c/r 0 = 1.21. Therefore, the measiured
mean square fluctuation level was 3.2 times larger than that
predicted by the renormalized theories, i.e.,
'ro
(.5f(l),5f(2))~=3.2 (2r ,D'12
d(f(l)) 3(f(2))
av
4V
(46)
Note that the enhancement factor of 3.2 is more acci irate
than the approximate solutions given by Eqs. (40) and (43)
might imply, since we have used only the ratio of the ;olutions.
Figure 15 shows the result of the orbit tracking neasurements of (Av 2 ) in the presence of the external w aves
only. We made the measurements in the region |vi glh
(where df,/dv~~0) so that the production of clumps w ould
not interfere with the analysis. The mean square velc city
change increased linearly with time, consistent with par ticle
diffusion [cf. Fig. 15(a)]. Figure 15(b) shows the result!s for
the mean square relative velocity change of particle pairs,
whose initial separation was x- = 0.1 2A , v =0.0,lVh.
The results for an initial separation of x = 1.47AD,
v. = 0.02v,, are shown in Fig. 15(c). As in the decay turbulence case, the results are consistent with the predictioris of
the orbit stochastic instability model. Actually, this is a case
where the model should most unambiguously apply, i.e ., in
an externally applied turbulent wave spectrum. It is c-lear
E
A
V
0
30
(Op
240
I0
b
A
V
10
Wpt
12
24 0
FIG. 15. Measurements of a part icle's
mean square velocity change ((A4,2))
and the mean square relative ve] ocity
change between two particles ((A v2 ))
for the driven case without self-c(onsistency (a) normalized (4V2 ) for
80<wpt<240; (b) normalized (4 V )
for initial separation x-/ D = 0.12,
-/v, = 0.02; and (c) norma lized
(4V2 ) for initial separation x_/
A = 1.4, v/vh =0.02.
C
A
' and
that D _<D for t - 80o7 '<re,(x_,v _)~40ow
kox_~x_/AD
*l.D- 2D for t-80w7' >,r1 . For
kox_>1, D_ ::2D for all t>80ar'.
Comparison of Figs. 11 and 15 show that the orbit exponentiation times r,(x_,v_) are comparable for the selfconsistent and non-self-consistent field cases. We note that
ko and D (cf. Figs. 11 and 15) are also comparable in the two
cases. Therefore, even though one case involves the plasma
self-consistent fields while the other does not, they are essentially indistinguishable as far as orbit stochastic instability
[ra,(x_,v_)] is concerned. However, as we have seen, the
mean square fluctuation level in the driven run is affected by
the plasma self-consistent fields [cf. Eq. (46)]. This result is
consistent with the results of the decay turbulence experiments. An individual fluctuation decays at the rate
r, (x-,v-) in accordance with the orbit stochastic instability model of T12 in Eq. (7). However, the aggregate fluctuation level decays at a much slower rate. We believe that,
though an individual fluctuation decays at the rate
r.,(x_,v) -', the mean square (aggregate) fluctuation level
decays more slowly because of the tendency for fluctuations
to recombine.
V. HOLE MODEL AND INTERPRETATION
In this section, we show that the essential features of the
simulation results can be understood in terms of a collection
of Bernstein-Greene-Kruskal-like holes. An isolated phasespace hole is a Bernstein-Greene-Kruskal equilibrium,5"'
and has been shown to be a state of maximum entropy subject to constraints of mass, momentum and energy.' In an
ion-electron plasma with immobile ions (as in these simulations), an electron phase-space hole is self-binding. The
phase-space density surrounding the hole, being of opposite
charge from the hole, is attracted to the region of depleted
charge." Alternatively, we note that the holes have negative
mass so that two phase-space holes will attract each other.5
For an isolated, self-bound hole in equilibrium, the hole
depth ( -f), and the hole potential must be sufficient to trap
(bind) the hole. If the hole is Ax by Av in width, then an
approximate calculation based on the maximum entropy argument yields5
f=Av(6w2 )'g(x/)',
g(z) = (1 + 2/z)[I - exp( - z)] - 2
A -2 =2
240
"'p t
2449
Phys. Fluids, Vol. 26, No. 9, September 1983
PV
du(v -u)-
(f)9
(P.Y. means principal value and A-+A D as v-+O). As Ax/A
approaches 0 or oo, g has the limits - (4x/A )2/6 and - 1
respectively. At v = 0 of the Maxwellian distributions,
(f) =v;'(21r)- 2 =fo, so that Eq. (47) becomes
fo
80
(48)
and
'
V
(47)
where
(2
2
7r)
4v
6v,,
4
-
(50)
2D-
For Av~vh and AX~~AD, Eq. (50) implies an equilibrium
hole depth f I<o.
Berman, Tetreault, and Dupree
2449
Of course, turbulent fluctuations cannot be exact equilibria since they are continuously colliding or interacting with
each other. Berk et aL.have investigated some of the properties of binary collisions between two phase-space holes." In
a collision between two holes with small relative velocity, the
two holes can merge into a single-hoje structure (cf. Fig. 6 of
Ref. 11). For larger relative velocities, tidal deformations of
the holes occur (cf. Fig. 7 of Ref. 11). They explain the tendency of holes to coalesce by a useful gravitational analogy
in which holes may be regarded as gravitating masses. In
Ref. 5, hole coalesence is shown to be the result of the plasma's tendency to attain states ofhigher entropy. The entropy
is increased as the phase-space density is mixed during the
hole-hole coalescence. In this way, the "coarse-grained"
average of f (and thus, the electrostatic fields) is reduced. We
would expect both hole coalescence and tidal deformation to
occur simultaneously in a turbulent plasma. We therefore
model turbulent fluctuations as a collection of interacting
holes with the interaction consisting of two competing processes-the coalescing of holes with the concomitant increase of Ax and A v scales and the breaking up of the holes
into smaller fragments due to D_ diffusion.
The tendency of holes to coalesce and increase their Ax
and Av scales can be understood5 by considering the hole
mass M, energy To, and entropy - o-. When two holes (1)
and (2) coalesce that have a nonzero relative velocity, the
self-energy of the resulting hole (3) is always less than the
sum of the original self-energies, i.e., T 03<TO + T 2 . The
total mass, however, is conserved (M3 = M, + M 2 ). For
holes with Ax/A < 1, the entropy - o-is an increasing function of Ax/A which in turn is an increasing function of M 2/
To. Therefore, the holes with Ax/A < 1 tend to coalesce since
this will lead to increased M 2 /T 0 , Ax/A, and - a. Furthermore, one can also show that for Ax/A < 1,
(A V/Vt )2 = (Ax/A ) M(nmA ) ', so that Av also increases as
the holes coalesce. On the other hand, the entropy (for constant TO) decreases with Ax/A for Ax/A> 1 and is proportional to (TOM )1/2. For this case, the most probable final state
for interacting holes is to concentrate all the energy in a hole
of length Ax/A =4 (with a corresponding mass of M 2 /T=5)
and put the remaining mass in a hole of zero energy, i.e., mix
it into the positive bf background. This means that hole coalescing will produce larger Ax scales but not greater than a
few A D. However, Av could increase without limit. Since fis
determined by Ax and Av [cf. Eq. (50)], the fact that the x and v- widths of C(x_,v_) remain approximately constant
as C (x-,v-) decreases with time implies that f for an individual fluctuation remains constant but that the packing
fraction of the fluctuation must decrease.
For comparisons between the observations and the hole
model, it is convenient to express Eq. (50) in terms of the hole
area A = AxA v and the parameter r = Ax/A v. Therefore,
__air
-- =
6
/
A
1\/2
g/A
\I/2
-
A
(51)
where Ax = (-A )M2,Av = (A /r)1/2, andAD
2DD is the
area of a Debye-length-long hole. If we identify r with the
characteristic scale ratio AX/A v defined in Sec. IV, then we
find that r= 10w,-' at w, t = 120. With this value of r,
2450
Phys. Fluids, Vol. 26, No. 9, September 1983
A D = 0.IA D vh. The independence of r can be understood
by the following intuitive argument. Consider a steady-state
distribution of holes. As the holes collide with each other,
"tidal" electric field effects will limit the scales that can coexist simultaneously in the system. Coexistence of disparate
scales demands that holes with different Ax, or A v, do not
destroy each other by tidal forces. The electric field of hole
(1) must be comparable to the self-binding field of another
hole (2) across which it acts, i.e.,
I_
or
(
'x
AVu '
~
1~d)
\Ax),
_
A
~kAx
dx7/2
Ax 2
'
(52)
(53)
/2
In order to obtain Eq. (53), we have assumed Ax<A D and
used Poisson's equation dE /dx~A
zVf~(A v/Ax) 2 . We would
not expect this approximate constancy of r for holes with
A >A D. For example, any large holes produced by coalescing
will have Ax on the order of a few Debye lengths but arbitrarily larger Av. Indeed, we recall from Sec. IV that
Ax/Av<5w-' for the large holes at w t = 220. However, it
appears that r-~10-' is approximately valid for the
smaller holes which satisfy A D >A
, where A, = Axdv,
is given by
f(Ax,,Av,) = -fo .
(54)
Equation (54) describes the deepest holes in the systemthose holes where all the particles have been removed. Any
fluctuations with Ax<Ax, and Av<,Av, will be unbound debris, since f will not be deep enough to bind. With
so that
we find that A,=0.l001AoVh
r = 10a)-,
Ax,=0.lAl andAvj=O.Olvh.
The initial checkerboard hole fluctuations had
Of= ±fO, Ax = AD, and Av = 0.lv,,. Equation (50) implies
that these holes were overbound (fo> If I) by a factor of 10.
Figure 4 for the time decay of the correlation function shows
that the system remedied this by tearing up the initial A D
holes down to scales of approximately A I where
f= -fo = f(A ). This decay process took approximately
60e,- '. Subsequently, the correlation function increased in
width. Apparently, approximate Bernstein-Greene-Kruskal-like holes formed by 60w,- 'and their subsequent coalesence leads to the production of holes with larger A. In this
regard, we recall that (see Sec. IV) the measured correlation
function appears to be composed of two v _regions. The core
of small v_ region of C(0,v_) is of size A <A I = 0.001UD vth
and persists throughout the simulation (cf. Figs. 4 and 5). We
interpret the A = A, boundary of the core to be due to
f(A) = -fo holes that were formed from remnants of the
initial checkerboard fluctuations. The A (A, core is due to
the smallest A holes (A =A ) and, in principle, any unbound
debris. However, we show in Sec. VII that the unbound debris makes little contribution to the core. The production of
the v |>(A 1/,r)1/2 width of C(0,v_ ) subsequent to W, t = 60
is due (we believe) to the coalesence of the smallest (A = A 1)
holes into larger hole structures. This increase in hole size
beyond A =A , is apparently limited to the velocity scales
Berman, Tetreault, and Dupree
2450
only. Here C (x ,0) retains its form during 60co, t 220, implying a smooth distribution of spatial scales whose maxi.These features of C (x-,v -) are consistent with
mum is ::
the maximum hole entropy arguments of Ref. 4. Fluctuation
energy tends to flow into holes with larger and larger velocity scales. The most probable holes, however, are a few Debye lengths long.
As we have seen in the decay experiments, the production of large v_ scale fluctuations occurs even as the mean
square fluctuation level decays. We interpret this fact as the
result of the tendency of holes to coalesce. As two holes coalesce, hole and interstitial material between the holes mix so
that the coarse-grained phase-space density is reduced. Alternatively, we can view this as a reduction in the hole-packing fraction due to hole coalescing. The driving force behind
this coalescing and mixing process is the tendency of the
holes to attract each other or self-bind. It is also this tendency which explains the slow decay rate of the aggregate fluctuation level. As we have seen from the tilt of the correlation
function contours and the orbit tracking experiments for
p = i, an individual fluctuation decays at the rate
7 '(x _,v_ : as the holes collide with each other, their constituent particles undergo stochastic instability. However,
the hole fragments of &f<0material resulting from these
collisions tend to recombine (because of the self-binding ten-
dency) into new holes composed of different (or unrelated)
hole fragments. The self-binding or recombination effect is
somewhat less than the r,7 '(x _,v _) collision rate. For p = 1,
there is a net decay rate of order of [3,r, (x-,v )] -' (cf.item
B 1 of Sec. I).
The tendency of the bf<0 hole fluctuations to attract
each other and coalesce also provides a simple interpretation
of the P ( Sf) observation in the decay experiments. Consider the t = 0 curve of Fig. 2(c). As time elapses, the background material tends to repel itself and mix with the hole
material. Alternatively, the reduction in the number of holes
(due to hole coalesence) creates more room in phase space
into which the background material can expand. The production of holes with different phase-space area [and according to Eq. (51), different f] will broaden the initial P ( &f >0)
peak. Equivalently, as the holes tend to coalesce into new
holes, the mixing of hole and background material reduces
the coarse-grained phase-space density V/. The rate at
which the mixing reduces the 6f <0 fluctuations is less than
that for Sf >0 since the holes tend to self-bind. These considerations imply a distribution P( bf) such as that shown in
Figs. 8(a) and 1(c).
As we discussed in Sec. I, one requirement for a strong
negatively skewed P( 6f) is that the hole-packing fraction
(p) be much less than J. Equivalently, fluctuations with TV
large and negative must be more likely than 6f large and
positive. Clearly, this will occur in the decay experiments if
the holes tend to self-bind. If there is an external source of
clumps-as in the driven experiments-the hole-packing
fraction will be maintained at one-half. Though any individual fluctuation will tend to decay (and reduce the hole-packing fraction), new clumps are being created by the rearrangement of fo(v) throughout phase space. This explains why we
2451
Phys. Fluids, Vol. 26, No. 9, September 1983
observed P ( 6f ) to be Gaussian in the driven experimentsboth with and without self-consistent fields.
VI. A THEORETICAL MODEL OF P( Wf)
As we have shown in Sec. V, the essential features of the
observations can be explained by a simple model of interacting Bernstein-Greene-Kruskal-like holes. Although the
discussion of Sec. V is sufficient to understand the qualitative features of the model, it is important to determine if the
model can explain any of the quantitative aspects of the simulation results. Therefore, we have developed a mathematical model for P( 6f ). As we shall see, the P( 6f ) model we
propose has many simplifying assumptions. However, reasonable agreement is obtained with the observations (cf.Sec.
VII).
Recall that P ( 6f ) was obtained by making a histogram
of the particle number fluctuation in a phase-space window
of area A.. As we have shown above, the P( 6) observations are consistent with a distribution of interacting Bernstein-Green-Kruskal-like phase-space holes. We will assume that each hole is identified by the area A in which it fits.
Because of hole coalescence and decay, it seems reasonable
to assume a continuous distribution of hole sizes A. Therefore, the theoretical model ofP ( 6/) that we propose has two
essential ingredients: the number (itwill be negative) of particles, N (A ), in a hole of area A; and the number of holes,
F(A )dA, between A and A + dA in a window of size A.. We
define 9( bf,h) as the probability of observing a fluctuation
6f in a window that has 1i holes in it on the average. Using
Nh (A) and F, we solve a model equation that relates
9(f,h)
to the average number of holes in a window
[h = fdA F(A )]. we then adjust h so that the solution
9( ffi) agrees with the observed 9( 6/). The idea is to
model P( f ) with a distribution of holes (F)where each hole
has a specific phase-space structure [N (A )]. We assume that
there is a smallest (A) and largest (A.) hole size at any given
time so that h. = fj'dA F(A ). We therefore model P ( 6/)
with the probability 9( 6fji.) of observing the fluctuation
6f in a window that has f. holes in it on the average.
Lacking a detailed theory, we are forced to make intuitive and simplifying assumptions about the expressions for
Nh (A ) and F(A ) in a turbulent plasma. We first consider the
case where the window area is (much) larger than any of the
holes (A . >A ). We will ignore the possibility (unimportant if
A >A ) that the window overlaps only a portion of a hole.
Then, N (A ) = /(A ) A, where - f(A ) is the depth of a hole
of area A. In order to obtain a model of f(A ) for a turbulent
plasma, we resort to a modified version of the isolated hole
expression (cf. Sec. V). This seems to be a reasonable approach since the effects of diffusion and hole binding (coalescence) are comparable, i.e., the aggregate fluctuation lifetime
is of order of three times the lifetime of an individual fluctuation. First, we will assume that small (Ax<A .) holes have
the same ratio of Ax/Av. This assumption is motivated by
the simulation results as discussed in the previous section.
We therefore consider Eq. (51) with r = 10w'. A second
modification of the isolated hole expression (50) is also required. We note that Eq. (50) describes an approximate hole
Berman, Tetreault, and Dupree
2451
in which all the hole mass is concentrated within an area
A = AxAv and a uniform depth - ? However an actual
hole of the same mass with Ax<A D will be more spread out
with a Ax and Au approximately twice as large. Therefore,
the average fover the larger area will be about half as large
as that given by Eq. (50). On the other hand, when Ax>A D I
Eq. (50) is reasonably accurate but the simulation results
indicate that r = Ax/Au ~ 2a,- ' compared with its value of
l0I 7' for Ax<A D. Both the large and small Ax/A D limits
are reproduced by the expression
f
f,
[FaA\\
/ aA 1/2
copr KAD)
g LD)
dir 1
=
6
1
where a::6,r= l0mc,', and A D = O.,hA
-
,
D
(55)
is the area of
the Debye-length-long hole. A simple analytical expression
for f/fo can be written which agrees well with Eq. (55) in the
limit of large (A >A ) and small (A<A D) hole dimensions:
f..
fo
I
or
fAD
(I
(56)
).
F(A)dA=(A./A)(p/A)dA,
(57)
where p depends on A in general. Equation (57) implies that
holes with larger A are fewer in number than those with
small A. If p(A ) is a slowly varying function of A, then Eq.
(57) has a simple physical meaning. Using Eq. (57), the sum
of the areas for the holes with areas between A and eA (e is
Euler's constant e = 2.718...) is A,.,., =AAp(A ) so that
p(A ) = A .. ,/A.. Therefore, p(A ) can be interpreted as the
fractional phase-space area of a window that is occupied by
the holes of area between A and eA. We refer to p(A ) as the
hole-packing fraction. If p(A ) is independent of A, then all
holes have the same packing fraction. We point out that this
would imply that large holes would be composed of small
holes. We assume that p(A ) = p is independent of A. With
this restriction, we use Eq. (57) to calculate the average number n(A ) of holes with areas A or less in a window of size A..
If A I is the area of the smallest hole, then
dA 'F(A ') = n, l - Al),
(58)
where n, = pA./A is the average number of holes of areaA,
in the window. The maximum value that n takes is
n = n.-- pA./Am where Am is the area of the largest-size
hole.
9( bf,R) can be related to the average number of holes
in the window n. To obtain this relation, we first consider the
simplified case where the phase space is populated with
holes-all of which have the same area. If these holes are
randomly distributed, the probability of observing n holes in
a window, when there are n holes per window on the average,
is
9 (n,) = (n"/n!) e-"
(59)
This distribution is skewed unless n -R>8,
2452
ni)= 30(n,5) -
~l
q(n -
l,h).
(60)
Equation (60) can be readily generalized to a distribution of
different hole sizes. For this, we define -4 to be the number
of particles per window that have been removed from the
holes (in that window). We also define (NBG ) as the average
number of particles in a window due to the f >0 background material. Clearly, (N) = (NBG) + (-4). We also
define h = h(A ) to be the average number of holes of size A or
less in the window. Then, the probability 9[fi(A)] of
finding .4' particles removed from the holes in a particular
window by counting only holes of size A or less, satisfies
A
1I/2
We will find it useful to use Eq. (56) instead of Eq. (55) for
analytical calculations in Sec. VI.
In addition to Nh (A ), we also require a model equation
for the hole distribution F(A ). For this, we consider the simple expression
R(A ) =
distribution
Gaussian
the
9(n,h) approximates
exp[ - (n - n) 2 /(2(6n 2 ))]. Because the holes are randomly
2
) = ((n - li)2 ) = h. The sindistributed in the window, (On
gle hole size distribution (59) satisfies the following difference-differential equation
in which case,
Phys. Fluids, Vol. 26, No. 9, September 1983
0 P(Jf)=9(A-,R)- 9 [-I-- N, A),f
dan
.
(61)
We can derive Eq. (61) by relating the probabilities for two
distributions that differ only by a small set of holes with
+ 4) can be written as the
A < 1. The probability q(Ifi
sum of two independent probabilities:
g(x'f, +An)
= 60(f,) (1 - Ah) + A R
(,41 -
.
N,)
(62)
The first term in Eq. (62) is the probability of observing the
original hole distribution with no additional holes contained
in the set AR. The second term is the probability of observing
one hole in the set A R multiplied by the probability of correspondingly observing Nh fewer particles (one hole less) in the
original distribution. Equation (62) is just Eq. (61) expanded
to lowest order inAR4 1. In order to obtain 9 ( f ,;i,) from
Eq. (61)-which we then identify with the observed P( i5f)we note that f = (N - (N))/A. = (X - (X))/A,. In
(h),,)A
addition, we require Nh(h) = Nh [A () I = f[A (
which, with Eqs. (56) and (58), can be written as
Nh (ii
1/
-D
=f0A.( , 7T)-1
\A"
_n
-
,
(63)
I
for A <A D.
In general, Eq. (61) is difficult to solve. As in the simplified case of Eq. (60), 9(XR) of Eq. (61) is very skewed and
sensitive to variations in n when n is small. Moreover, the
singular behavior of N (n) [cf. Eq. (63) as n-+n,] makes
9(f,)
of Eq. (61) sensitive to h as n-+R,. As we shall see,
Rn)
however, there is an intermediate region of R where 9(
of Eq. (61) is only weakly skewed and thereby changes slowly
with R. For this reason, we choose to solve Eq. (61) as an
initial value problem in h where the initial value n, lies within this intermediate region of R. We can write the formal
solution to Eq. (61) as
'9f,5)
= e
+ e
(Xni)
mi
M=
M!a,
dn...
a
dhm
X 9[-V - N(h) . -- N, (m,i ] ,
Berman, Tetreault, and Dupree
(64)
2452
Of
where fi = fij + A h. The average value of .P is
(>)=f
,i) =
(.
di
(65)
d'Nh(;'),
whereas the mean square value is
(,,A
2
f
) =
dA(
,
)2
-
s
dh' Nh (i') 2 .
=
(66)
Note that (A-) and (&A"4) are defined as functions of h.
In order to evaluate Eq. (64), we consider the process of
adding holes with increasing A [i.e., h(A )] into the window.
First, we note that if dNh (R)/d = 0, then Eq. (61) reduces to
Eq. (60) and .(.,i)
will be Gaussian for R>8. Therefore,
we suspect that 9 (,5i)
of Eq. (61) will be Gaussian if
8
dlnN,,(i)
(<1,
tn
-4
d
' =1
luation of Eq. (64). We choose the initial value h, to be an
intermediate value of h where the condition Eq. (67) begins
to be violated or equivalently where s(h) is at its minimum.
Let this value of R be denoted by fg*
(Vfl,) is approximately a Gaussian and 9(AR,.) can be written as
(_
--
3 9P(Xi)
,
(68)
where
f
'
(67)
0f-dfr
f-
2
b
FIG. 16. Theoretical skewness s(Pi) for the model discussed in Sec. VI.
and h>8. In order to verify this, we consider the skewness
s(R)
'
0
_1.9 (V, R).
(71)
p~,-,) +,d 9 ,
,)=
e -h
where A 9 is the skewed part of
,
Here A 9 need
only contain the m = 1 term in the series Eq. (64), since
A = ,. - fi 41. Therefore,
(69)
Using Eq. (64), we find that
fy dh'N
N3(')
s(R) = [ d
(70)
[ Oi dh' N ih )]u
Figure 16 is a plot of Eq. (70) using N,, (h) obtained from Eq.
(55). If R is small (few holes in the window), the skewness is
large because R (8. For larger fi, the skewness decreases and
depends only weakly on fi. In this intermediate region of fi,
Eq. (67) is satisfied. If h is increased further so that Eq. (67) is
violated, then the skewness increases again. This latter increase of skewness is rapid and occurs in a region 4h< 1. We
are therefore, led to the following prescription for the evaexp[
-
(.'
-
(.))
2
A
=
d-0 [.IV - N(),f,].
The Gaussian 9 (Af~i)
=exp[
( -1
12)2/2 (
is due to those holes with area
A (A, and are, on the average, h, = h(A,)> 8 in number. We
can evaluate .(A'i) by splitting up the hole number distribution (for fiihg) into regions of width Mn>8. Then,
9( RI)will be Gaussian in each region if Eq. (67) is satisfied. We can obtain 9 (1,rig) by combining (convolving) all
such regions together. For example, combining two regions
together, we obtain
/2.5I NI(i)]
exp[
-
(Xf-
[ 21-6h 2 N (& 1 1/ 2
p6
(72)
'[721r32 N
2)2/22 NW2I
)2
] 1/2
42) 121
(3
(21r(&/f/2),2) /2
where
(0)12
+
N
65
2
(
=
2 ).
()
1+ (0)2
and (&/142) 12 = 6hN h((,)
Convolving all regions leads to the result
= (21r(&12))-1/2e
g(Ihg)
( _(X2 _ (
Xy)
)
2
,2
(74)
In order to compare the theoretical model (.,,,)
with the observations, we note that the P( 6]) curves are
plotted as a function of
f = (N - (N))/A. =6N/A.
= (.Af - (.))/A.-We therefore rewrite Eq. (74) as
Xexp [ - (&N+ A(N)) 2 /(2(6N
where the mean square value of X for the Gaussian is
(of'g2
(75)
dNh()
and the mean value can be written as
(fig
(')g
= ()
+
J2,
2453
Phys. Fluids, Vol. 26, No. 9, September 1983
where
- (0))=
we
have
(76)
used
_4"-(
2
)g)I,
(77)
)g =6N-((X)g
SN - A(N), where
A(N) = - I
Rl
dh' N (R').
-
1/2
9(N,fii) =(21r(,N 2
e++
d N,( i)
(78)
is the shift of the Gaussian part of 9(&N,j)from the origin
Berman, Tetreault, and Dupree
2453
6N= 0. Note that from Eq. (66), (N 2 ), is given by Eq. (75).
Note also that if A 9 = 0 (no skewness), then A (N ) = 0 so
that 9(6N,,, ) would be an unshifted Gaussian. Qualitatively, the skewness-producing holes will shift the Gaussian
to the right (bN>0) of the origin and enhance (stretch) the
5N(O tail of the Gaussian. Using the conservation of total
probability, we can calculate the ratio r of the area under the
"tail" portion A 9 (8N,5z) to the area under the Gaussian
portion 9(N,i.). We find
r =AR,
(79)
for A h<1. Once r is known, Eq. (79) becomes a relation for
A R. Here A H = ii - f, so that Eq. (58) gives an additional
determination of AR
Ai =p(A/A,
- A./Am).
(80)
We note that Eq. (80) is more model-dependent than Eq. (79),
since r of Eq. (79) can be obtained directly from the measured
P (Wf) curves.
In the interest of simplicity, we have up to this point
restricted the model 9(6N,hm ) to holes with areas less than
the window area. We can now relax this restriction by redefining N, (A ) and F(A ) so that they include holes with arbitrary A. IfA >A, the number of holes with area between A
and A + dA that contribute to the window is pA ' dA.
Therefore, we redefine F(A ) to be
F(A )dA= ( + A/A ) (p/A ) dA.
(81)
We point out that this modified version of Eq. (57) is probably only qualitatively correct. In particular, it still suffers
from the assumption that p(A ) = p = const over the entire
range of A. Using Eq. (81), we find that the average number
of holes that the window sees that have area less than or
equal to A is
;!(A)=n,(l -A/A)+pln(A
/A),
(82)
where n, =pA./A,. The correction to Eq. (58) for A>A.
holes will be small if n,(1-A1/A)>plnA /AI or
A./A,>InA m/A1I for A>A 1. This is satisfied for all but the
smallest windows. We redefine Nh (A ) as the number of particles in a hole that contribute to the window of area A.. We
neglect again the possibility that the hole and window overlap. For A>A., this is a serious omission-especially for
holes whose shape differs significantly from that of the window. However, since the A (A D holes are more numerous
than those with A >A D and have Ax/A v = T, we should apply the 9 ( Sfi.)
model to windows that have
Ax./AV, = r. This will tend to minimize the occurrence of
hole-window overlap. With this restriction in mind, we
write the following (very) approximate expression for N1 (A)
which includes holes with arbitrary A:
Nh(A )=f(A )AA./(A
+ A.).
Phys. Fluids, Vol. 26, No. 9, September 1983
l
q = 2p[
1/2 -
+ A1 (Am
A,
For A
<A
D
\
A)
)1/2
A,1/2
A1 Ag
AD Q)
(84)
and A, <AD, Eq. (84) can be simplified to
g=, 2 P A."
(
+ P
,
(85)
where we used Eq. (58) to express Ag in terms of n, -fi,
- n, - 5 + M -AR +pA /A.. Similarly, the normalized mean square width of the Gaussian, u = (.N2
(83)
Equation (83) correctly gives the number of particles in a
hole that contribute to the window for the limiting cases of
A>A. and A<A..
Equations (71), (72), and (77) can be evaluated in conjunction with Eqs. (55), (75), (78), and (83) to yield an approximate solution for q( bf,R.). As we have seen, the solution procedure assumes that any skewness in 6 ( Zf,5m)
2454
will occur only in the small region Ah = fm - fg 41 beyond
R.. Though this assumption is strongly supported by Eq. (70)
for the skewness (cf. Fig. 16), we have attempted to verify the
approximation procedure directly. We have, therefore,
solved Eq. (61) explicitly as an initial value problem. We
chose the window Ax. = 1.227A D, A v. = 0.125Vh so that
Ax./Av. = r 10O- '. Using a packing fraction
p = 0.1494, Al/A D = 1.329 x 10-2, and Am/A D = I (see
Sec. VII), we calculated from Eq. (70) that the minimum
(absolute) value of the skewness s(h) occurred for
ng = 16.930. Using Eqs. (75), (77), and (78), we then obtained
the initial value .9(bf,,ig) for Eq. (61). Equation (61) was
then solved numerically for R>R9 . The result for R. - n,
= h = 0.249 is shown in Fig. 17. The distribution clearly
exhibits the characteristic features of the observed P( f)
[cf. Fig. 8(a)]. The measured skewness of Fig. 17 is
s = - 1.05 and agrees well with Eq. (70). We also find that
choosing the initial condition such that 80ii<R, led to
f( 6f,h) that was approximately Gaussian until >f..
Moreover, the skewness of .( bf,R) was small and weakly
dependent on h in the region 8<R<F *,.
This result is consistent with Eq. (70) (and Fig. 16) and justifies the procedure of
constructing Eq. (77) by convolution. The small values ofA R
justify the retention ofonly the m = 1 term of Eq. (64) for the
approximate expression Eq. (72) for A 9.
We have used the 6( 6f, .)model to calculate some
of the prominent features of the observed P ( 8f). In doing
so, we have used Eq. (55) for numerical calculation and the
approximate expression (56) for analytical work. The shift
and mean square width are readily identifiable features of
the P( 6f) curves. We can easily calculate these quantities
from the 9 ( Sf_,.) model. Since theP( 6f) curves are plotted against Sf/fo, we calculate the normalized shift
q = A(N)/(foA.). Using Eqs. (78), (83), (58), and (56), we
obtain
FIG. 17. Theoretical P( 3f) generated by
numerical solution of 9(h,5) equation.
C.
-0.4 -0.2
0
02
04
R/ fo
Berman, Tetreault, and Dupree
2454
(f' A '),
follows from Eq. (75),
c (A.) =
dA
p
(86)
__
f ,
A ±A.
Using Eq. (54) and the approximate analytical expres sion
(56), Eq. (86) becomes
rA
o (Aw)=p - Lln 1 +A./A
A/A
g
++ A,D
In Ag +A
AD)
A +A.J
+ A,2
Ak
)2(A
1
\
1)
(87)
or
o' (A.)~=p
A,1 + A.",
In A
(88)
A 1 'Ag +Aw
when A, <Ag <A D. Similarly, the mean square width olfthe
full (8N,5.) distribution is
f(A))
r2A) A+.p
2
(89)
daA + A. \ fo
.
This is just Eq. (86) with A, replaced by Am. The first ter min
o, is due to holes of size A <A D, while the other terms ci
ome
from holes with A >AD. The scaling o2 (A)-A;
for
A,,>A, and A w >A is consistent with a random distr
intion of holes with sizes A <Aw, i.e., (SN2 )- (N). The s
metrized correlation function Eq. (35) for the one-param yieter
hole model (cf. Ref. 13) is
02.(4.
=
1
C(Ax.,Av.) = C(A.) = 4
d2
92
-
Axi2w
(SN 2).
(90)
d
For A. =A, =4x,Av,<A,, (6N 2 )
so that we
C(A,)A
can write the peak value of the correlation function as
C (A,)
)
~ o. (A,) ~ p
f2
,
[ I+ +-("-
A,
A,)
+ 2 In-
7,)i
(91)
For A, <A D, C (A,) is due mainly to the deepest, smal[lest
holes, i.e., those for which f(A )::f(A)= -fo.Largerh oles
contribute to the width of the correlation function. For
Am >A, and A,A D (A., theAm and A, scale lengths in .the
logarithm factors of o (Am) indicate that C (A .) will qua litatively be composed of two regions: one with width :A, due
to deep holes, and one of width :Am due to holes of size
VII. COMPARISON OF P(
OBSERVATIONS
W
) MODEL WITH
In order to compare the hole model of P ( Sf) with the
decay turbulence observations, we consider the typical curve
shown in Fig. 8(a) (4x. = 1. 2 3A D, Av. = 0.11 72 vh,
w, t = 120). We calculate from Eq. (37) that discrete-particle
fluctuations do not contribute significantly to the P( Sf) of
Fig. 8(a).
Using the measured values o02(A.)=0.0137 and
o4,, (A,) = 0.18 from Fig. 8(a), we solve the exact Eqs. (89) and
(55) for o'2 (A.) and o (A,)as simultaneous integral equations for p and A . We find that p = 0.18 and Am = 1.7A ,
= 1. IA,. "Completing the Gaussian" in Fig. 8(a), we find
2455
that the ratio of areas for the tail to the completed Gaussian
is r = 0.26. Equation (79)then gives h = 0.26. We note that
this value of 4Ai for this window is consistent with the numerical solution of Eq. (61) discussed in Sec. VI. With this
value of Ah, we use Eq. (80) to obtain Ag = 0.43A. or
Av = 0.654v.. Then, Eq. (89) for o(A.) allows us to compute og = 0.08. The measured value from Fig. 8(a) is ~0.07 .
Using Eq. (84) and the values Ah, Ag, and Am just inferred
above, we find that q = 0.03 whereas the measured value
from Fig. 8(a) is 0.05. At opt = 120, we use the values
p = 0.18 andA. = 1.7A D to find that Eq. (90) is consistent
with the qualitative shape of the correlation function. These
points of approximate agreement show that the hole model is
reasonably successful in describing the "low-order" moments of P { Sf) represented by q and o2 (A,).
The high-order moments of P( f) are more difficult to
describe by the
( Sf,um) model. For example, the skewness is very sensitive to 4h and, therefore, to p and Am. This
is to be expected from Eqs. (63) and (70) since
m - n =pA./A. (cf. Fig. 16). Indeed, in the solution of
( bf$ ) depicted in Fig. 17, we found that in the region of
n near n,, a small change (0.2%) in h would produce a 40%
change in the skewness. Therefore, using the values of
p = 0.149 and Am = A, (which are close to the values
p = 0.18 andA ' = 1.7A D inferred from a2.), we were able
to obtain reasonable qualitative agreement between P( bf)
of Fig. 8(a) and the 60 ( bf,ui.) solution in Fig. 17. In particular, a skewness of order - 1 is easily obtained for Ah~~0. 2 .
However, using the values of p = 0.18 and A. = 1.7A ,
Eq. (70) gives a skewness of - 0.2 and rather - 1. If we
repeat this calculation for the window of Fig. 8(a) for all a, t
[i.e., obtain p(t) andA, (t)from the simultaneous solution of
Eq. (89) and Eq. (91) and then use Eq. (70)], we would obtain
the solution in Fig. 8(b). It is clear that the model and measured skewness do not agree too well. Given the sensitivity of
s(h) to fi, it is better to first choose p and A. so that the
skewnesses agree. Comparison between Figs. 8(a) and 17
shows that this prescription also gives reasonable agreement
between the model and observed values of q and o.2
The higher-order moments of P ( bf) are also particularly sensitive to the large A holes. This sensitivity occurs because f
d Sf f "l9(Sf ,,,,)-fI" dn-[N, (fi)
Phys. Fluids, Vol. 26, No. 9, September 1983
0
{ X
~X X
x
s
+ + +
X
+
- 14
0
W220
FIG. 18. Measured skewness ( + ) and theoretical skewness (X) in time.
Berman, Tetreault, and Dupree
2455
0
-Q80r
+
+
+I
,4 _
+
s
220
0
Wp t
FIG. 21. Theoretical values of the hole-packing fraction p(t) inferred from
or' (A,) and ot,(A.) ( + ); the dashed line is a fit for Eq. (96).
3
AW/AD
FIG. 19. Theoretical skewness s(A.) versus window area A..
= ff,"dAF(A)[N(A)]n, where Nh-f-k
for holes
with A >A D [cf. Eq. (55)]. This helps to explain the fact that
the model and observed skewness agree better in Fig. 18
(where A. <AD) than in Fig. 8(b) (where A.>AD). The approximate model is more accurate in treating the holes with
A <AD. We note from Fig. 8(a) (where A. >A D), that the
value bf/f 0 = - 0.4 implies a hole with Av=0.4 vh and
Ax~ AD [see Eq. (50)]. We recall that this value of Ax/
4v~2.5w, is consistent with the value r(5.3ca- 'inferred
from the v_ tail of C(0,v_) (cf. Sec. III). These large holes
imply a value for Am that is larger than the one we obtained
from orn above. This discrepancy can be attributed to the
fact that the width of P ( 6f) is less sensitive to the tail of
P ( bf) (and therefore, to holes with large A ) than the higherorder moments such as the skewness and, therefore, predicts
a smaller value of A,.,
A property of the 9( bf,H) model that is less sensitive
to the hole distribution is the scaling with window size. Using windows where Ax/4v.~=r= l0ao,7,
and
A IA<A. (AD, we have observed that the normalized shift q
and width of the completed Gaussian a, decrease approximately as A 1/2. This is in agreement with Eqs. (84) and (88)
where we note that Af>pAw/Am for these windows. The
model skewness [derived from Eqs. (55), (70), (83), and (58)]
decreases with A. as is shown in Fig. 19. The scaling is seen
to be in qualitative agreement with the measured skewness
for windows where Ax./v.~ 10=7
l
and A. >A,/ 2 (cf.
Fig. 9). The initial increase in the skewness of Fig. 9 is due to
the particle discreteness (see Sec. III).
The scaling of the 9(^f8f,fim) model with time can be
easily obtained for the case A, <A. <A,, AR >pA ./A., and
Am =A,. Equations (85H91) then imply that q', o2(Aj,
o,(A), and org(A,) should scale in time with the packing
fraction p(t). The measured values for these quantities (normalized to their value at w, t = 120) are shown in Fig. 20 for
the window Ax. = 0.6A DA vw = 0.05v,,. The values of p(t)
plotted in Fig. 20 were obtained by solving o. (A.) and
oa,2(A,) [Eq. (89) and Eq. (55)] simultaneously for p(t) and
Am(t) (cf. Fig. 21). The quantities q2 , o (A,), o (A.), and
oa(Ag) appear to decay in time with p(t) in agreement with
the model. Figure 18 shows that the scaling of the skewness
with time agrees with the model for w, t> 80, i.e., when the
hole model is valid. The skewness agreement is not due to
p(t) alone. Though Nh -p (so that s-p -1/2), s also depends
onA (t ).This explains why the skewness varies linearly with
time from Fig. 18 whereas p(t ) - 1/2~t /2 from Fig. 21(a):s(t)
increases faster than p(t )a/2~t
/2 becauseA. (t) increases
with time. We also note that a, (A.) appears to decay with
p(t) alone in Fig. 20, even though ai (A.) depends on Am (t)
as well as p(t). Again, this is consistent with the fact that the
lower-order moments of - ( 3f',f) (such as o2) are less sensitive to the large holes (Am) than the higher moments (such
as s).
0
4.0
+
+
-
a
A
oA
E
+
-
0
4.
+
n
50
240
4.
Wpt
220
0
FIG. 20. Time dependence of various moments ofP ( 6f!) inferred from the
theoretical model discussed in Sec. VI: the normalized quantities q2(A.)
(dot), oa.(A,) (times), o(A.) (circle), oa2(A.) (box), and p (triangle) for a window where Aw = (0.6AD) (0.05V,h).
2456
+
++
Phys. Fluids, Vol. 26, No. 9, September 1983
Wpt
FIG. 22. Theoretical values of the phase-space area A, (t) of the largest hole
inferred from o4 (A,) and c4(A.).
Berman, Tetreault, and Dupree
2456
following equation for the packing fraction:
2
FIG. 23. P( 6f ) for a small window at
w>
t = 220.
Remnants of deep holes
initially present persist.
-1.0
__0
0.8
8f/f 0
Reference toA ,(t ) in Fig. 22 shows that the first tei
Eq. (91) (i.e., the 1) dominates the term proportional t(
A D throughout the simulation. It is for this reason that
in Fig. 21 closely approximates C (Ax ,,A v) displayed iii
6. We conclude that the unbound debris [which is not ini
ed in Eq. (91)] makes little contribution to the corret
function. This is consistent with the rapid mixing of i
fluctuations into the background plasma, i.e., the unb<
debris should have a mixing rate described by Eq. (15):
little or no recombination should occur for these fli
ations. Therefore, the core region of C(x -,v-) is due ti
internal structure of the holes of area A,. The width o
core is constant in time since the deepest holes always nave
A =A,, i.e., f(A 1)= -fo. That these holes exist eve n at
w, t = 220 is clear from Figs. 1(a) and 23. However, the significant skewness of Fig. 23 implies that these deepest 1ioles
are few in number. This is consistent with the time dec,ay of
the peak of the correlation function.
VIII. A MODEL OF FLUCTUATION DECAY
A simple but intuitively appealing model can be constructed for the time behavior of fluctuations during thi- decay experiments. We begin by recalling the hole model ii iterpretation for the correlation function depicted in Fig. 1. The
core region (A <A) is due to the deepest holes [i.e., t.hose
satisfying f(A ) = -fo] whereas the main body (A I A ( AD )
is due to larger holes formed by the coalescing of the Jeep
holes. This interpretation of the peak of the correlation f anction is consistent with Eq. (91) since A,/A
10-2 and A.
D throughout the experiment, i.e.,
C (AS)::::pf(A j)2,
(92)
where we have used Eq. (56) for the deepest holes. These Jeep
holes maintain their characteristic scale sizes (A >A) eve,nas
their mean square fluctuation level decreases. This is e,asily
verified by comparing the core region of C(x_,v_ ) at
,, t = 60, 120, and 220. If the scale sizes remain appi oximately constant in time, Eq. (50) implies that their de,pths
remain constant also (cf. Fig. 23). Therefore, their n iean
square fluctuation level decays because their packing fraction decays, i.e.,
dln (bf 2 ) _ dlnp
(93)
dt
dt
Figure 21 shows that the holes are closely packecI at
p t = 60, i.e., p=0.4 The holes will interact strongly with
each other: hole-hole collisions (D-) will lead to the brea kup
of holes into fragments and to the recombination (coadescence) of the fragments into new holes. The resulting de-cay
of the mean square fluctuation level can be modeled by the
2457
Phys. Fluids, Vol. 26, No. 9, September 1983
dp
dt
-
(94)
Vp.
The quantity v in Eq. (94) is a phenomenological decay rate.
If the holes are closely packed (pz), then vzA v/Ax, where
Ax and Av are the approximate hole dimensions. More appropriate to a turbulent system of holes where p =1 (e.g.,
the
driven
experiments)
is
the
identification
v~[reAx,Av)]-'~~r-'. In either case, we have v~ 10w-'
since r- '~Av/Ax~ 10w,- as discussed in previous sections. However, as the holes are destroyed in the decay experiments, the collision frequency between holes will decrease because the phase-space separation between the
remaining holes will increase, i.e, the hole-packing fraction
will decrease. Therefore, the hole-hole collision frequency is
pA v/Ax. However, hole fragment recombination will reduce this hole-hole collision rate by the factor b -' where
b> 1. Consequently, the decay rate of the aggregate fluctuation (hole) level is
v = 2p [b (Ax/Av)]~.
(95)
We stress that the decay rate is determined by both the decrease of the packing fraction in time and the tendency of the
holes to recombine. Solving Eqs. (93)-(95) for t>to gives
(96)
p(t) = p(to)/[l + 0.2a,(t - to) p (to)b -1] .
Presumably, we must choose t> 80co,- ' since it is only after
this time that Bernstein-Greene-Kruskal-like holes have
formed. Equation (96)fits the data well if b:::2.4 (cf. Fig. 21).
We note that this value of b implies that the effects of hole
recombination (self-binding) and turbulent decay (collisions
between holes) are comparable. Note also that when p = 1,
the net decay rate of (2.4r,)- is approximately the same as
the result of the driven experiments (cf.Sec. VI and item B 1
of Sec. I).
It is interesting to use the model described by Eqs. (93)(95) in an effort to rectify the discrepancy between the existing clump theory and the observations of Ref. 6. In Ref. 6, an
instability was observed in a one-dimensional plasma simultion in which an electron Maxwellian distribution [ fo
= (21rv)~112 exp[ - (v - vD) 2 /(2v,)] drifts with a velocity VD relative to an ion Maxwellian distribution [ fe
2
= (21rv?) - /2 exp [ - vA/(27)] . The observed growth rate
2
of (6f ) for the ions as a function of vD/v, is shown in Fig.
2
24. The corresponding linear growth rate (i.e., y) is also
shown. (Note that yL was shown in Fig. 4 of Ref. 6.) The
observed instability has been identified with the clump instability.7 The instability of clumps in a one-dimensional electron-ion plasma has been investigated theoretically in Ref. 7
(where the point of marginal stability is computed) and in
Ref. 8 (where the growth rate is calculated). If one neglects
the self-binding feature of the clumps, such calculations predict a higher stability threshold and a lower growth rate than
those observed in the simulations. It has been suggested that
these discrepancies would be eliminated if self-binding were
included in the theory.
In order to see how this might be done, let us first consider the theory of electro.9-ion clumps in the absence of selfbinding. The electron clumps are described by an equation of
Berman, Tetreault,
and Dupree
2457
O.10 0 r
and the turbulence length scale k ,~' satisfies
x
2
2
YL/wpe
y/wpe
-- I
2= 02Re
N''(k)k 2
A (k).
'f) dk \k| 1 - 2arlk
IN'e(k )/k,
(104)
Because of Eqs. (102) and (103), (R ie)2 = R eeR " in Eq.
(101)[cf. Eq. (100)]. In order to obtain the growth rate Y from
Eq. (101), we must solve Eqs. (102) and (104) simultaneously.
r
This can be done numerically. The point of marginal stabil0.0 01,,
ity follows from 2R ' = 1 [cf. Eq. (101) with y = 0] and Eq.
5
VD/Vi
FIG. 24. Theoretical growth rate of the nonlinear clump instability when
hole self-binding is included (2y) and linear (2 yrL) growth rate of (6f') as a
function of vD/vi.
(104). For the parameters used in the simulation of Ref. 6, the
numerical solution of these coupled equations gives a threshold drift of VD ~2.5v, at v, = vi. However, the instability
threshold in the simulation was observed at v, ~1.5v,. Note
that the linearstability boundary occurs at UD = 3.924v, for
the parameters in Ref. 6.
the form
(I
+
\
(97)
G'= S',
i.e., Eq. (1) where T 1 '-+24, in accord with the stochastic
instability model, G' = (6f efe) is the electron fluctuation
correlation function, and S' is the electron clump source
term. As in Eq. (7), S' is composed of two terms. It is shown
in Ref. 8 that Eq. (97) can be written schematically as
G =
G'
+
Ge,
+
-) G, _ R "
cl,
11,
+
li
6',
(100)
Detailed calculations show that the growth rate satisfies
(when y, <I and yr,< 1) 8
(1 + 2cre- R e) (I + 2cyro - R 'e)= (R )2 , (101)
where ro and rJ are Eq. (14) for the electrons and ions, respectively. For yre, <1, the parameter c 3 and reflects the
fact that rI (x _,v _)>ro. Here Rie, RCC, and R " are similar to
Eq. (24) and are defined as
N'e(k) = Im e(k,kv,) Im e'(k,kv,)
1T~e(k,kv,)12
2458
Phys. Fluids, Vol. 26, No. 9, September 1983
rate of v =(br,)-, where b = 3. Therefore, the simulation
results imply that Eqs. (98) and (99) should be written as
+
G
i= + R'
(105)
and
G' =R jN' + R i
+ a
(106)
\ T b, }
r,
The factor b does not appear on the right-hand side of Eq.
(105) and Eq. (106) because the R 's are proportional to re,.
Consequently, the growth rate expression Eq. (101) is replaced by
(1/b + 2cri-O - R-') (1/b + 2cri - R i)=(R
')2
(107)
The point of marginal stability now follows from
(1 + 2yri, - R)(1 + 2 yr, - R )= eeR.
dk|kj
Nab(k)
_
(k)
f
I - 2r~k IN"(k)/koH'
where A (k) is defined by Eq. (26),
rate of the mean square fluctuation level is less than rj7'. In
the driven experiments (where p = 1), we observed a decay
(99)
where R "~(f )2. For rl <1 and scale lengths less than the
typical clump dimension, GC~GC and G'~G'. If we now
set 8/dt-+2y, the simultaneous solutions of Eq. (98) and Eq.
(99) yield
Rab=
due to D_ and ballistic streaming. However, the results of
the driven and decay experiments imply that the net decay
(98)
where R- and ReC are the two-species analogs of the parameterR defined in Eq. (24)ofSec. II. The R" term in Eq. (98) is
due to the diffusion term in s [cf. the first term of Eq. (8)].
The ion electric fields rearrange the electron phase-space
density and create electron clumps: D(fi )2 (E 2
e,[f)2
G'(f, )2/y,~R
"G'/4,. The R" term in Eq. (98) is due
to the second term in S' and transfers momentum between
electrons and ions, i.e., R 'e~ -f; fi. The ion clumps satisfy an equation analogous to Eq. (98):
at
We can include the effects of self-binding in the clump
theory in the following way. First, we recall that in Eqs. (98)
and (99), the r.7 ' term describes the decay of fluctuations
(102)
(103)
2R'e = 1/b .
(108)
Using the value b ~ 3 inferred from both the decay and driven experiments, we expect the stability threshold to occur
when 2Re ~0. 3 3 and Eq. (104) are satisfied simultaneously.
Numerical solutions show that this occurs (for v+ = v,)
when VD ~ 1.5v,, in agreement with the observations of Ref.
6. Figure 24 shows the growth rate observed in Ref. 6 and the
growth rate y calculated from Eq. (107). In accord with Ref.
6, we used ro = o(m,/m,) 2 = 0.5r:'::40ac, '. The parameter c was chosen to fit the data at vD = 2.5v,, i.e., c = 2.9.
(Note that this value is consistent with the theoretical prediction for c.) We have used v, = v, for all drift values except
VD = v, and vD = 1.5v,, where, in accord with the measurements of Ref. 6 we used v+ = 0.5v1 . Equation (107) reproduces the essential features of the data for all but the larger
drifts. However, we note that the yr' (1 and yrl < 1 restrictions assumed in the derivation of Eq. (101) are violated for
these larger drifts. Moreover, Eq. (101) excludes the existence of linear plasma waves that would become nonlinearly
Berman, Tetreault, and Dupree
2458
unstable (via resonance broadening) at these larger values of
VD (note that linear stability occurs at VD = 3.924v). Note
also that we have assumed that the effect of self-binding on
Eqs. (105) and (106) is independent of VD. This may be incorrect. However, use of the approximate value b = 3 seems
appropriate near )D ~0 since for this drift, the two-species
plasma approximates the stable (one-species) plasma discussed in this paper. Indeed, if we had used b 1 for VD = 0,
thereby omitting self-binding, Eq. (107) would give
2
Y~ - 9.2 X 10~co -i rather than the value 2 y~ - 3.3
X 10' for b = 3. Another note of caution concerns the
neglect of collisions in Eq. (101). In principle, collisions can
lead to the destruction and production of clump fluctuations. The production of clumps results when discrete particle fluctuations rearrange the phase-space density [as in an
analogous fashion to D" in Eq. (17)]. Therefore, both S and
T, of Eq. (1) will be enhanced by collisions. The net effect of
these competing collisional corrections is difficult to calculate. However, estimates show that they are comparable at
the marginal point observed in the simulation of Ref. 6, i.e, at
v ~ 1.5v,. This conclusion would tend to justify the use of
Eq. (101) (where collisions are neglected) in interpreting the
marginal point observed in Ref. 6.
For p < 1, the clump instability picture is replaced by
one of isolated growing holes. A hole that is isolated in phase
space (i.e., p--.0), has been shown to be unstable in an ionelectron plasma with opposing (velocity space) density gradients.5 - For instance, an electron hole can exchange momentum with the ions that are reflected by the hole. As a result,
the hole moves up the density gradient, thereby getting deeper. This instability is the single (isolated) fluctuation analog
of the clump instability. We can see this by writing Eqs. (105)
and (106) in the limit of an isolated hole. The ions act coherently (yr,-+oo) so that only the momentum transfer term
(proportional to R") survives in Eq. (105). Also, for an isolated hole, p-*O, so that the -rT 1 fluctuation collision term
vanishes. The isolated electron hole grows at the rate
yh -((4/Ax) R '',
(109)
since -r,-+)e--*4x/Av for an isolated hole. The marginal
point for the isolated hole instability follows from Re = 0.
However, for a collection of holes, i.e., nonzero packing fraction, instability is more difficult to achieve. The source term
(R") must be large enough to overcome the decay of fluctuations due to hole-hole collisions. These collisions are more
2459
Phys. Fluids, Vol. 26, No. 9, September 1983
frequent for larger packing fractions. Therefore, for a collection of electron holes, we would expect a growth rate satisfying
(110)
.
e -2p
RAv
b Ax
Ax
Equation (110) follows from Eqs. (105) and (106) by making
the replacement r '--+p 4 v/Ax and noting that the r ' factors on the right-hand sides of Eqs. (105) and (106) are replaced by 4 v/Ax since the R 's are proportional to r,, . Moreover, if the ions act coherently (ri--+co), only the R'" term
survives in Eq. (105). Equation (110) implies that R ' = 2p/b
replaces Eq. (108) for the point of marginal stability. Therefore, as p--+O, the threshold VD for the onset of the instability
will approach zero. The effect of the fluctuation packing
fraction on the instability is now under study and will be
published elsewhere.
Note addedin proof: We have recently become aware of
two-dimensional computer simulations of fluid turbulence
(J. C. McWilliams, Workshop on Chaos and Coherent
Structures in Fluids, Plasmas, and Solids, Los Alamos, 1983)
where similar intermittent phenomena have been observed--namely, the emergence of isolated vortices and a
decrease in packing fraction generating intermittency.
ACKNOWLEDGMENTS
The use of Macsyma at M.I.T. for some calculations is
acknowledged.
This work was supported by the National Science
Foundation and the Department of Energy.
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